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Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

Microsoft  Corporation 


http://www.archive.org/details/elementsofplanesOOcandrich 


THE  ELEMENTS 


OF 


PLANE    AND    SOLID 


ANALYTIC    GEOMETRY 


BY 


ALBERT   L.    CANDY,  Ph.D. 

ASSISTANT  PROFESSOR  OF  MATHEMATICS  IN  THE  UNIVBRSITY 
OF  NEBRASKA 


BOSTON,  U.S.A. 

D.    C.    HEATH   &   CO.,   PUBLISHERS 

1904 


/:Z<MyC 


Copyright,  1904, 
By  D.  C.  Heath  &  Co. 


PEEFACE 

Analytic  Geometry  is  a  broader  subject  than  Conic  Sections.  It  is  far 
more  important  to  the  student  that  he  should  acquire  a  good  knowledge  of  the 
analytic  method,  that  he  should  comprehend  the  generality  of  its  processes,  and 
learn  how  to  interpret  its  results,  than  that  he  should  obtain  a  detailed  knowl- 
edge of  the  properties  of  any  particular  set  of  curves.  Furthermore,  there  is 
a  certain  interrelation,  or  interdependence,  between  the  various  branches  of 
elem-entary  mathematics.  Experience  in  teaching  these  subjects  has  convinced 
"me  that,  on  the  ground  of  expediency  alone,  this  interdependence  should  be 
recognized  in  the  class  room.  In  the  study  of  mathematics,  as  well  as  in  its 
applications,  Algebra  and  Geometry,  Analytics  and  Calculus,  are  mutually 
helpful.  Hence  these  branches  should  not  be  studied  entirely  apart.  As  all 
these  branches,  or  at  least  more  than  one,  must  finally  be  used  in  the  complete 
solution  of  many  problems,  tliere  seems  to  be  no  good  reason  why  the  student 
should  not  be  taught  to  do  this  as  soon  as  possible. 

For  these  reasons  a  fuller  treatment  than  usual  is  given  of  the  general  analytic 
method  before  taking  up  the  study  of  the  conic  sections,  and  subjects  have  been 
introduced  that  are  not  ordinarily  included  in  text-books  on  Analytic  Geometry. 
The  method  of  the  Differential  Calculus  is  the  only  way  of  studying  the  slope  of 
curves,  and  furnishes  the  best  means  for  finding  the  equation  of  the  tangent 
and  the  normal.  The  graphical  method  of  illustration  and  the  derivative  are 
indispensable  in  the  study  of  the  Theory  of  Equations.  The  use  of  the  Deriva- 
tive Curve  in  the  theory  of  equal  roots,  together  with  the  fact  that  the  ordinate 
of  the  derivative  curve  is  the  slope  of  the  Integral  Curve,  naturally  suggests  a 
possible  converse  relation,  and  leads  easily  and  logically  and  with  no  difficult 
transition  to  the  study  of  Quadrature  and  Maxima  and  Minima. 

It  is  believed  that  the  elementary  treatment  of  these  subjects  here  given  will 
tend  to  meet  the  needs  of  scientific  and  technical  students,  who  now  require  a 
knowledge  of  the  graphical  method  and  the  simpler  elements  of  the  Calculus  at 
the  earliest  possible  moment ;  and  that  it  will  also  be  helpful  to  the  general 
student  who  pursues  the  study  of  mathematics  no  farther.     Moreover,  in  the 


^«  i/AkJi  «>j^ -^  C  %  ^~V 


IV  PREFACE 

secant  method  of  finding  the  equation  of  the  tangent,  the  reasoning  is  essen- 
tially the  same  as  in  the  method  here  used,  but  the  student  seldom  or  never 
comprehends  its  significance.  And,  furthermore,  he  never  uses  the  method 
save  in  the  case  of  the  conic  sections,  whereas  the  derivative  method  is  one  that 
he  can  always  use. 

The  subjects  discussed  in  Chapter  VI  need  not  be  taken  at  the  time  or  in  the 
order  in  which  they  occur  in  the  book.  Or,  if  the  teacher  prefers  to  pursue  the 
old  established  method  of  teaching  each  branch  of  mathematics  exclusively,  he 
.  may,  at  his  discretion,  omit  this  entire  chapter  without  interfering  in  any  way 
with  the  continuity  of  his  course.  While  this  book  has  been  in  preparation, 
my  own  plan  has  been  (with  students  who  have  not  previously  had  the  Theory 
of  Equations)  to  give  in  substance  the  theorems  contained  in  §§  63-71  immedi- 
ately after  the  work  on  curve  tracing,  or  symmetry.  The  remainder  can  be 
given  any  time  after  Chapter  V  has  been  read. 

In  finding  the  equations  of  loci,  special  emphasis  is  given  to  the  meaning  of 
the  parameters  which  appear  in  the  final  equations,  and  the  significance  of  a 
variation  in  their  value,  and  a  full  discussion  and  a  thorough  geometric  inter- 
pretation of  the  result  are  rigidly  insisted  on  from  the  beginning.  The  teacher 
should  never  lose  sight  of  this  vital  principle. 

Polar  coordinates  and  their  relations  to  rectangular  coordinates  have  been 
introduced  at  the  very  beginning. 

The  conic  section  is  first  briefly  studied  geometrically.  Its  defining  property 
is  proved  in  this  way,  from  which  its  general  equation  is  shown  to  be  of  the 
second  degree.  The  two  central  conies  are  treated  simultaneously  by  using  the 
double  sign  in  the  standard  equation.  In  this  way  much  time  is  saved,  and 
the  similarities  of  the  properties  of  the  two  conies  are  presented  in  a  striking 
manner. 

As  the  book  is  intended  for  beginners,  numerous  illustrative  examples  are 
given  in  the  first  part  on  Plane  Geometry,  and  also  a  large  number  of  exercises. 
The  numerical  examples  have  all  been  prepared  especially  for  this  book.  An- 
swers are  given  to  only  a  few  of  these,  as  it  is  far  better  to  check  results  in  such 
exercises  by  constructing  an  accurate  figure.  A  unique  feature  in  the  way  of 
exercises  is  found  in  the  list  of  Miscellaneous  Problems  on  Loci  that  occur  in 
the  phenomena  of  everyday  life.  These  cover  a  wide  range  of  subjects  and 
should  be  of  interest  to  students  in  any  department.  The  study  of  mathematics 
should  not  only  develop  the  power  of  investigation,  but  should  also  cultivate  the 
habit  of  carefully  examining  interesting  phenomena.  I  hope  these  problems 
will  help  toward  the  accomplishment  of  these  ends,  and  at  the  same  time  tend 
to  bridge  over  the  chasm  between  the  theoretical  and  the  practical.     They  are 


PREFACE  V 

placed  at  the  end  of  Part  I,  so  that  they  may  be  assigned  at  any  time  without 
seeming  to  have  been  passed  over. 

The  theory  of  the  second  part  on  Solid  Geometry  is  somewhat  fuller,  and  the 
examples  are  considerably  more  extensive  both  as  to  number  and  character,  than 
is  usually  the  case  in  elementary  books.  The  chief  new  feature  that  has  been 
introduced  is  the  use  of  the  notion  of  Contour  Lines  in  the  tracing  of  surfaces. 
This  idea,  as  well  as  the  whole  subject  of  surface  tracing,  has  not  hitherto  been 
sufficiently  emphasized. 

Where  the  proof  in  Solid  Geometry  is  the  same  as  in  the  corresponding  propo- 
sition in  Plane  Geometry  the  demonstration  has  not  always  been  repeated.  In 
two  instances,  viz.  §  154  and  §  169,  an  entirely  different  method  of  proof  has 
been  used.  This  has  not  been  done  simply  for  the  sake  of  variety,  although 
this  would  be  a  sufficient  reason,  but  because  the  algebraic  results  obtained  in 
this  way  admit  of  a  much  broader  interpretation.  The  student  should  be  re- 
quired, as  an  exercise,  to  apply  these  methods  of  proof  to  the  corresponding 
propositions  in  Plane  Geometry,  and  vice  versa.  As  a  suggestion  to  this  end, 
appropriate  references  are  given  in  all  these  sections.  If  this  is  done,  the  student 
will  be  able  to  prove  for  himself  the  harmonic  properties  of  the  conic  section. 

I  have  put  two  small  sections,  I  and  II,  in  the  Appendix  rather  than  assign 
them  to  any  particular  place  in  the  body  of  the  text.  The  method  of  finding 
the  direction  of  a  curve  at  the  origin,  given  in  I,  I  have  found  to  be  helpful  as 
early  as  in  the  section  on  curve  plotting  in  Chapter  II.  If  used  at  all,  it  should 
at  least  precede  the  formal  study  of  slope. 

I  wish  to  thank  most  heartily  all  my  colleagues  in  this  university  who  have 
aided  me  so  kindly  in  the  work,  and  to  acknowledge  my  special  obligation  to 
Professor  Ellery  W.  Davis,  who,  from  the  inception  of  the  plan  to  the  completion 
of  the  book,  has  given  me  much  valuable  assistance.  I  am  also  much  indebted 
to  Professor  George  D.  Olds,  of  Amherst  College,  and  Professor  E.  V.  Hunting- 
ton, of  Harvard  University,  who  have  read  the  entire  manuscript  with  great 
care  and  offered  many  helpful  suggestions. 


A.  L.  C. 


Thk  University  of  Nebraska, 
May  25,  1904. 


CONTENTS 


PART  I.     PLANE   GEOMETRY 

CHAPTER  I 
Coordinates,  Distances,  and  Areas 

BF.CTIONB  PASS 

—  1-4.     Cartesian  coordinates.     Exercises 1 

6.     Polar  coordinates.     Exercises 6 

6.     Relations  between  rectangular  and  polar  coordinates.     Exercises  .  7 

--  7-9.     Distance  between  two  points 8 

10-13.     Areas  of  polygons 12 

Examples  on  Chapter  I 16 


CHAPTER  II 

Loci  and  their  Equations 

14-20.     Illustrative  examples  and  definitions 18 

21-22.     Plotting  loci  of  equations.     Exercises 23 

23.     Use  of  graphic  methods.     Illustrations 28 

24-26.     Intersection  of  loci.     Exercises 32 

27-29.     Symmetry  of  loci.     Exercises 34 

30-37.     Finding  the  equations  of  loci.     Exercises 38 


CHAPTER  III 

The  Straight  Line 

38-49.     Standard  forms  of  the  equations  in  rectangular  and  polar  coordi- 
nates.    Exercises     ..;......  46 

50-51.     Equations  of  the  straight  line  in  oblique  coordinates.     Exercises  .  62 

Examples  on  Chapter  III 63 


Vlll 


CONTENTS 


CHAPTER  IV 
Transformation  of  Coordinates 

BE0TI0N8 

63.     To  move  the  origin  to  tlie  point  (A,  k) 
54.     To  turn  rectangular  axes  through  an  angle  d 
Examples  on  Chapter  IV     . 


PAGE 

67 


CHAPTER  V 
Slope,  Tangents,  and  Normals 

55-56.  Examples  of  limiting  values  of  ratios.     Exercises 

58.  Geometric  meaning  of  the  derivative  of  the  function  /  (x) 

59.  Illustrative  examples  of  derivatives.     Exercises  . 
60-61.  General  formulae  for  differentiation     .... 

62.     Tangents  and  normals 

Examples  on  Chapter  V 


70 

72 
74 
76 
78 
78 


CHAPTER  VI 
Theory  of  Equations,  Quadrature,  and  Maxima  and  Minima 

63-80.    Theory  of  equations.     Exercises 82 

81.  Quadrature.     Exercises 101 

82.  Maxima  and  minima.     Exercises 105 


CHAPTER  VII 

Conic  Sections 

85.  Geometric  proof  of  the  defining  property  of  the  conic  section 

86.  Classification  of  the  conic  sections.     Exercises 

87.  General  equation  of  the  conic  sections 
88-91.  Standard  equations  of  the  conic  sections     . 
92-93.  Tangents      .         .         .         . 
94-95.  Pole  and  polar.     Exercises  .... 


113 
116 
117 
119 
126 
128 


96-100. 


CHAPTER  VIII 

The  Para-bola 

Properties  of  the  parabola  i/2  =  4  ax.     Exercises 
Examples  on  Chapter  VIII 


131 
136 


CONTENTS  IX 


CHAPTER  IX 
The  Circle 

8EOTION8  PA6B 

101-106.    Properties  of  the  circle.     Exercises 139 

Examples  on  Chapter  IX 146 


CHAPTER  X 

The  Ellipse  and  Hyperbola 

107-108,    Tangent,  polar,  and  normal 160 

109.     Geometric  properties.     Exercises 152 

110-111.     Conjugate  hyperbola,  and  director  circle 166 

112-114,     Auxiliary  circle  and  eccentric  angle.     Exercises  .         .        .  167 

115,     Asymptotes  —  definition  and  equations 162 

116-117,     Similar  and  coaxial  conies  —  diameters 164 

118-121.     Conjugate  diameters.     Exercises 168 

122-123.     Equation  referred  to  asymptotes,  and  polar  equation  .        .  174 

Examples  on  Chapter  X 176 


CHAPTER  XI 

General  Equation  op  the  Second  Degree 

124-126.     To  transform  the  general  equation  to  the  standard  forms   .        .  179 

Examples  on  Chapter  XI 187 

Examples  on  loci 188 

Miscellaneous  problems  on  loci 189 


PART  11.     SOLID   GEOMETRY 

CHAPTER   XII 
Systems  op  Coordinates,  the  Point,  Rectangular  Coordinates 

127-130.  Rectangular  coordinates.     Exercises 193 

131.  Polar  coordinates  and  direction  cosines        .        .        .        .         ,  197 

132.  Spherical  coordinates 198 

133.  Cylindrical  coordinates.     Exercises 199 


CONTENTS 


CHAPTER   XIII 
Loci 

BHCTION8  PAGK 

134-135.     The  locus  (in  space)  of  an  equation  is  a  surface  ....  200 

136-139.     To  trace  the  locus  of  an  equation.     Contour  lines.     Exercises    .  202 

140.  To  find  the  equation  of  a  locus 205 

141.  Surfaces  of  revolution.     Exercises 206 


CHAPTER  XIV 

The  Plane  and  the  Straight  Line 

142-143.     The  plane.     Exercises 209 

144-149.     The  straight  line.     Exercises 211 

150-151.     Transformation  of  coordinates 216 

Examples  on  Chapter  XIV 218 

CHAPTER  XV 

CONICOIDS 

152-165.     The  sphere.     Exercises 220 

156-157.     The  cone 224 

158.  The  ellipsoid 227 

159.  The  hyperboloid  of  one  sheet 229 

160.  The  asymptotic  cone  of  the  hyperboloid  of  one  sheet  .         .        .  231 

161.  The  hyperboloid  of  two  sheets 232 

162.  The  elliptic  paraboloid.         .        . 234 

163.  The  hyperbolic  paraboloid 235 

164.  The  paraboloids  the  limiting  forms  of  the  central  conicoids. 

Exercises 237 

165-168.     Tangent  planes  and  normals 238 

169-170.     Poles  and  polar  planes 241 

Examples  on  Chapter  XV 243 


APPENDIX 


I.     The  direction  of  a  curve  at  the  origin 245 

TI.     Example  illustrating  §  81 .246 

III.     Trigonometrical  formulse 248 


ANALYTIC  GEOMETRY 


CHAPTER   I 

COORDINATES,  LENGTHS  OF  LINES,  AND  AREAS  OF  POLYGONS 

Rectilinear  Coordinates 

1.  Let  X'X  and  F'  Y  be  two  fixed,  non-parallel  straight  lines,  in- 
tersecting in  the  point  O.  Let  P  be  any  point  in  the  plane  of  these 
lines.    Draw  HP  and  QP  parallel  to  X'X  and  Y'  Y  respectively. 


These  distances,  RP  and  QP,  determine  the  place  of  P  within  the 
angle  XOY.  That  is,  to  every  position  of  P  there  is  one  and  only 
one  pair  of  distances,  to  every  pair  of  distances  one  and  only  one 
position  of  P.  Moreover,  the  position  of  P  can  be  found  when  the 
lengths  of  the  lines  RP  and  QP  are  given,  and  vice  versa. 

Suppose,  for  example,  that  we  are  given  EP  =  a,  QP=  b,  we  need 
only  measure  OQ  =  a  and  OR  =  h  and  draw  the  parallels  RP  and 
QP,  which  will  intersect  in  the  required  point. 

2.  The  two  lines  RP  and  QP,  ot  OQ  and  OR,  which  thus  de- 
termine the  position  of  the  point  P  with  reference  to  the  lines 


2  COORDINATES  [3 

X'X  and  T^Y  are  called  the  Rectilinear  or  Cartesian''^  Coordinates  of 
the  point  F.  QP  is  called  the  Ordinate  of  the  point  P,  and  is  denoted 
by  the  letter  y;  RP,  or  its  equal  OQ,  the  intercept  cut  off  by  the 
ordinate,  is  called  the  Abscissa,  and  is  denoted  by  the  letter  x. 

The  fixed  lines  X'X  and  F'F  are  called  the  Axes  of  Coordinates, 
and  their  point  of  intersection  0  is  called  the  Origin.  When  the 
angle  between  the  axes  of  coordinates  is  oblique,  the  axes,  and  also 
the  coordinates,  are  said  to  be  Oblique ;  when  the  angle  between  the 
axes  is  right,  the  axes  and  the  coordinates  are  said  to  be  Rectangular. 

If  OQ  =  a  and  OR  =  6,  then  at  P,  x±=a  and  y  =  h;  at  Q,  x  =  a 
and  2/  =  0 ;  at  i?,  «  =  0  and  y  =  h',  and  at  0,  cc  =  0  and  2/  =  0. 

The  axis  XX  is  called  the  Axis  of  Abscissas,  or  the  x-axis;  and 
F'  F  is  called  the  Axis  of  Ordinates  or  the  y-axis. 

3.  Let  OQ  and  OQ'  be  equal  in  magnitude  to  a,  and  let  OR  and 
OR'  be  equal  in  magnitude  to  h.  Through  Q,  Q',  R,  and  R'  draw 
lines  parallel  to  the  axes,  and  intersecting  in  Pj,  P2,  P3,  P4. 


Jy' 

Now  at  all  of  these  four  points  x=a,  in  magnitude,  and  y  =h, 
in  magnitude.     Hence  in  order  that  the  equations  x  =  a  and  y=^h 

*  This  method  of  determining  the  position  of  a  point  in  a  plane  is  due  to  the  French 
philosopher  and  mathematician,  Descartes.  Hence  the  name  Cartesian.  The  new 
method  was  first  published  in  1637. 

"  It  is  frequently  stated  that  Descartes  was  the  first  to  apply  algebra  to  geometry. 
This  statement  is  inaccurate,  for  Vieta  and  others  had  done  this  before  him.  Even 
the  Arabs  sometimes  used  algebra  in  connection  with  geometry.  The  new  step  that 
Descartes  did  take  was  the  introduction  into  geometry  of  an  analytical  method  based 


4]  COORDINATES  8 

shall  determine  only  one  point,  it  is  not  sufficient  to  know  the  lengths 
of  a  and  6,  we  must  also  know  the  directions  in  which  they  are 
measured. 

In  order  to  indicate  the  directions  of  lines  we  adopt  the  rule  that 
opposite  directions  shall  he  indicated  by  opposite  signs.  It  is  agreed,  as 
in  Trigonometry,  that  distances  measured  in  the  directions  OX  (or 
to  the  right)  and  OY  (or  upwards)  shall  be  considered  jxysitive. 
Hence  distances  measured  in  the  directions  OX'  (or  to  the  left)  and 
0  Y'  (or  downwards)  must  be  considered  negative.  Therefore  (assum- 
ing a  and  b  to  be  positive  numbers) 

at  P„  a;  =  a,  2/  =  & ;  at  Pg,  a;  =  —  a,  2/  =  6 ; 
at  P3,  ic  =  —  (X,  y  =  —  6 ;  3i>t  P^,  x  =  a,  y  =  —b. 

Thus  the  four  points  are  easily  and  clearly  distinguished,  for  no 
two  pairs  of  values  of  x  and  y  are  the  same. 

If  all  possible  values,  positive  and  negative,  be  given  to  x  and 
to  y,  i.e.,  if  both  x  and  y  be  made  to  vary  independently  from 
—  00  to  4-00,  all  points  in  the  plane  will  be  obtained.  Moreover, 
to  each  pair  of  values  of  x  and  y  there  corresponds,  in  all  the  plane, 
one  and  only  one  point ;  to  each  point,  one  and  only  one  pair  of  values. 

4.  For  the  sake  of  brevity,  a  point  is  represented  by  writing 
its  coordinates  within  a  parenthesis,  the  abscissa  being  always 
written  Jirst.  Thus,  in  the  preceding  figure.  Pi,  Pg,  P3,  P4,  are  the 
points  (a,  b),  (—a,  b),  (—a,  —b),  (a,  —b),  respectively.  In  general, 
the  point  whose  coordinates  are  a;  and  y  is  called  the  point  (x,  y). 

When  the  axes  are  rectangular  it  is  convenient  to  distinguish  the 
parts  into  which  the  axes  divide  the  plane  as  first,  second,  third,  and 
fourth  quadrants,  as  in  Trigonometry. 

Because  of  simplicity  in  formulae  and  equations,  it  is  generally 
more  convenient  to  use  rectangular  axes. 

on  the  notion  of  variables  and  constants,  which  enabled  him  to  represent  curves  by 
algebraic  equations.  In  the  Greek  geometry,  the  idea  of  motion  was  wanting,  but  with 
Descartes  it  became  a  very  fruitful  conception.  By  him  a  point  on  a  plane  was  deter- 
mined in  position  by  its  distances  from  two  fixed  right  lines  or  axes.  These  distances 
varied  with  every  change  of  position  in  the  point.  This  geometric  idea  of  coordinate 
representation,  together  with  the  algebraic  idea  of  two  variables  in  one  equation  hav- 
ing an  indefinite  number  of  simultaneous  values,  furnished  a  method  for  the  study  of 
loci,  which  is  admirable  for  the  generality  of  its  solutions.  Thus  the  entire  conic 
sections  of  Apollonius  is  wrapped  up  and  contained  in  a  single  equation  of  the  second 
degree."    [A  History  of  Mathematics  by  Florian  Cajori,  p.  185.] 


4  COORDINATES  [4 

Accordingly,  throughout  this  book,  except  when  the  contrary  is 
expressly  stated,  the  axes  may  be  assumed  rectangular. 

EXAMPLES 

1.  In  what  quadrants  must  a  point  lie  if  its  coordinates  have  the  same  sign  ? 
different  signs  ? 

2.  Locate  the  points  (1,-  3),  (-  2,  4),  (5,  0),  (-  1,  -  3),  (4,  2),  (0,  3). 

3.  Construct  the  triangle  whose  vertices  are  the  points  (0,  4),  (—5,  —  1), 
and  (4,  -  3). 

4.  Construct  the  triangle  whose  vertices  are  (4,  —  1),  (1,  2),  (-1,  -  3). 

6.  Construct  the  quadrilateral  whose  vertices  are  the  points  (3,  4),  (  —  1,  4), 
(— 1,  —2),  (3,  —2).  What  kind  of  a  quadrilateral  is  it?  Consider  both 
oblique  and  rectangular  axes. 

6.  Plot  the  points  (8,  0),  (5,  4),  (0,  4),  (-  3,  0),  (0,  -  4),  (5,  -  4),  and  con- 
nect them  by  straight  lines.     What  kind  of  a  figure  do  these  six  lines  enclose  ? 

7.  P  is  the  point  (x,  y) ;  Pi,  P2,  P3  are  its  symmetrical  points  with  respect 
to  the  X-axis,  y-axis,  and  origin,  respectively.  What  are  the  coordinates  of 
Pi,  P2,  Ps  ? 

8.  The  side  of  a  square  is  2a.  What  are  the  coordinates  of  its  vertices  when 
the  diagonals  are  the  axes  ? 

9.  The  side  of  an  equilateral  triangle  is  2a.  What  are  the  coordinates  of  its 
vertices,  if  one  vertex  is  at  the  origin  and  one  side  coincides  with  the  x-axis  ? 

10.  Where  may  a  point  be  if  its  abscissa  is  2  ?  if  its  ordinate  is  —  3  ? 

11.  Can  a  point  move  and  yet  always  satisfy  the  condition  x  =  0?  y  z=0  ? 
both  the  conditions  x  =  0  and  y  =  0? 

12.  How  must  a  point  move  so  as  to  satisfy  the  condition  x=c?  y  =  d?  both 
these  conditions,  c  being  a  negative  and  d  a  positive  number  ? 

13.  If  a  point  moves  along  either  of  the  bisectors  of  the  angles  between  the 
axes,  what  is  the  relation  between  its  coordinates  ? 

14.  Where  may  a  point  be  if  its  coordinates  satisfy  the  condition  x^  +  y^  =:  a^? 
What  is  the  relation  between  the  coordinates  of  a  point  which  moves  so  that  its 
distance  from  the  origin  is  always  2  ? 

16.  If  a  line  AB  is  two  units  to  the  left  of  the  y-axis,  what  are  the  coordinates 
of  a  point  whose  distance  from  AB  is  three  units  ? 

16.  If  P  be  any  point  on  the  bisector  of  the  angle  between  the  ?/-axis  and  a 
line  three  units  above  the  x-axis,  what  is  the  general  relation  between  the 
coordinates  of  P  ? 


5]  COORDINATES  5 

PoLAK  Coordinates 

5.  Let  0  be  a  fixed  point  called  the  Pole,  and  OX  a  fixed  line 
called  the  Initial  Line. 

Take  any  other  point  P  in  the  plane  and  draw  OP.  The  position 
of  the  point  P  with  reference  to  the  line  OX  is  known  when  the 
distance  OP  and  the  angle  XOP  are  given. 

The  line  OP  is  called  the  Radius  Vector  of  the  point  P,  and  will  be 
denoted  by  p ;  the  angle  XOP,  which  the  radius  vector  makes  with 
the  initial  line,  is  called  the  Vectorial  Angle  of  the  point  P,  and  will 
be  denoted  by  ^.  .p 


Then  p  and  6  are  the  Polar  Coordinates*  of  P;  that  is,  P  is  the 
point  (p,  6).  As  in  Trigonometry,  it  is  agreed  that  the  angle  0  shall 
be  positive  when  measured  from  OX  counter  clockwise ;  that  p  shall 
be  positive  when  measured  in  the  direction  of  the  terminal  line  of 
the  vectorial  angle  0. 

In  determining  the  position  of  a  point  whose  polar  coordinates  are 
given  the  following  direction  will  be  useful :  Suppose  I  stand  at  0 
facing  in  the  direction  of  OX  To  get  to  the  point  (p,  6),  I  turn 
through  the  angle  6  to  the  left  or  right  according  as  6  is  positive  or 
negative,  then,  keeping  my  new  facing,  I. go  a  distance  p  forward  or 
backiuard  according  as  p  \^  positive  or  negative. '\ 

♦  Whenever  the  position  of  a  point  in  a  plane  is  determined  by  any  two  magnitudes 
whatever,  these  two  magnitudes  are  the  coordinates  of  the  point.  Tims  there  may  be 
an  indefinite  number  of  systems  of  coordinates.  For  an  explanation  of  other  systems 
which  are  in  common  use  see  Chap.  I  of  Elements  of  Analytical  Geometry  by  Briot  and 
Bouquet,  translated  by  J.  H.  Boyd. 

t  This  method  of  locating  points  by  means  of  coordinates  is  not  altogether  new  to 
the  student,  neither  is  it  confined  to  mathematics.  For  example,  when  we  locate  places 
on  the  surface  of  the  earth  by  means  of  their  latitude  and  longitude,  we  make  use  of  a 
system  of  rectangular  coordinates  in  which  the  axes  are  the  equator  and  some  chosen 
meridian.  When  we  say  the  city  B  is  forty  miles  north-east  of  the  city  A,  we  locate  B 
with  reference  to  A  by  means  of  a  system  of  polar  coordinates  in  which  the  initial  line 
is  tlie  meridian  through  A,  and  A  is  the  polo.  Let  the  student  suggest  other  familiar 
examples,  if  possible.    How  are  places  located  in  cities?  in  AVashington,  D.C. ? 


6  COORDINATES  [5 

EXAMPLES 

Plot  on  one  diagram  the  following  points  : 
1.    (4,30°),     (-3,135°),    (3,120°),    (-4,-30°). 
*  2.    (5,45°),     (-4,120°),    (3,-150°),     (-6,-240°). 

3.  (a,  iTr),  (-a, -J-tt),  (a,  -fTr),  (2a,  -fTr),  (-!«,  -^tt),  (a,  0),  (2rt,  tt). 

4.  (5,  tan-15),    (-2,  tan-i2),    (3,  -tan-i3),    (- 4,  tan-i  -  1). 

5.  ,(«,  tan-12),  (a,  -tan-i3),  (-a,  tan-if),  (-a,  -  tan-i  f), 
[a,  tan-i(-4)]. 

6.  Plot  the  points  (-6,30°),  (2,150°),  (2,  -90°)  and  connect  them  by 
straight  lines.     What  kind  of  a  figure  do  these  lines  enclose  ? 

7.  Plot  the  points  (a,  00°),  (&,  150°),  (a,  240°),  (&,  -  30°),  and  join  them  by 
straight  lines.    What  kind  of  a  figure  do  these  lines  enclose  ? 

8.  Find  the  polar  coordinates  of  the  vertices  of  a  square  whose  angular 
points  in  rectangular  coordinates  are  (3,-1),  (-1,  -  1),  (-  1,  3),  (3,  3). 

9.  The  side  of  an  equilateral  triangle  is  2a.  If  one  vertex  is  at  the  pole, 
and  one  side  coincides  with  the  initial  line,  what  are  the  polar  coordinates  of  its 
vertices  ?  of  the  middle  points  of  the  sides  ? 

10.  Change  "  equilateral  triangle  "  to  "  square  "  in  Ex.  9. 

11.  Change  "  equilateral  triangle  "  to  "  regular  hexagon  "  in  Ex.  9. 

12.  How  must  p  and  d  vary  in  order  to  obtain  all  points  in  the  plane  ? 
(See  §  3.) 

13.  Show  that  to  each  pair  of  values  of  p  and  6  there  corresponds  in  all  the 
plane  one  and  only  one  point. 

14.  Show  by  plotting  the  four  points,   (3,60°),    (-3,240°),    (3,  -300°), 
(  —  3,  —120°),  that  the  converse  of  Ex.  13  is  not  true. 

15.  Show  that  in  general  the  same  point  is  given  by  each  of  the  four  pairs  of 
polar  coordinates, 

(p,0),   (-p,7r  +  ^),   [p,  _(27r-^)],   [-P,  -(tt-^)]. 

16.  Show  that  for  all  integral  values  of  n  the  same  point  (p,  6)  is  also  given  by 

(p,  e  ±  2mr)  and  [- p,  6  ±  (2w  +  l)7r]. 

17.  Where  does  the  point  (p,  (9)  lie  if  ^  =  0  ?    if  ^  =  tt  ?    if  p  =  2  ? 

18.  How  can  the  point  (p,  6)  move  ii  6  =  cc?  it  p  =  a?  where  a  and  a  are 
constants  ? 

19.  What  condition  must  p  and  6  satisfy  if  the  point  (p,  6)  moves  along  a  line 
perpendicular  to  the  initial  line  ?    parallel  to  the  initial  line  ? 

20.  What  is  the  position  of  the  point  (p,  0)  if  p  =  a  cos  ^  ?    p  =  a  sin  d  ? 


6] 


COORDINATES 


Relations  between  Rectangular  and  Polar  Coordinates 

6.  Let  P  be  any  point  whose  rectangular  coordinates  are  x  and  y, 
and  whose  polar  coordinates,  referred  to  O  as  pole  and  OX  as  initial 
line,  are  p  and  6. 


T 

x' 

a    r 

S 

X 

'/ 

O 

y 

/ 

P 

/ 

Y^ 

Draw  PQ  perpendicular  to  OX. 

Then,  according  to  the  preceding  definitions, 

Oq  =  x,    QP  =  y,    OP=p,    zxop=e. 
From  the  right  triangle  PQO  we  have 

0Q=:  OP  cos  XOP    and     QP=OPsmXOP. 
:.sc  =  p  cos  0. 1 
y  =  psme.  t 


(1) 


These  equations  (1)  express  the  rectangular  coordinates  in  terms 
of  the  polar  coordinates. 

From  equations  (1)  we  find  the  corresponding  equations  express- 
ing the  polar  coordinates  in  terms  of  the  rectangular  coordinates  to 

be  ^ .,        ^ 

p  =  Vic2  +  2/2,  e  =  tan-i^ 


3C 


sine  = 


V 


cos  e  = 


iC 


.  (2) 

Va;2  +  2,2'  ^"'"      Vic2  4.yi' 

By  means  of  formulae  (1)  and  (2)  equations  in  either  system  of 
coordinates  can  be  changed  into  the  other  system  of  coordinates. 
It  is  seldom  necessary,  however,  to  use  equations  (2). 


DISTANCES 


EXAMPLES 

1.   Change  the  equation  p2  :=  (fi  cos  2  ^  to  rectangular  coordinates. 
Multiplying  the  equation  by  p^^  and  putting  cos  2  ^  =  cos^  d  —  sin^  6  gives 

p4  =  a2  (p2  cos2  ^  -  p2  sin2  61) . 
Whence  by  substituting  equations  (1)  we  have 

(X2  +  2/2)2  -  052(a;2  _y2). 

Change  to  polar  coordinates  the  equations 


2.  x2  4-  y2  —  2  rx.     Ans.  p  =  2  r  cos  6. 
4.   (2  x2  +  2  7/2  -  ax)2  =  a\x'^  +  2/2). 
Transform  to  rectangular  coordinates 


5.  p2  sin  2  ^  =  2  a2.    Ans.  xy  =  a^.      6.  p^  cos  J  ^ 


3.   x2-?/2 


Ans.  p2  =  0^2  sec  2  ^. 
Ans.  p^  =  a^  cos  i  ^. 

Ans.  ?/2  +  4  ax  =  4  a2. 


Distance  between  Two  Points 

7.    To  j^/i(Z  the  distance  between  two  points  whose  rectilinear  coordi- 
nates are  given. 

Let  P^ix^j  ?/i)  and  P2(^2?  2/2)  be  the  given  points,  and  let  the  axes  be 
inclined  at  an  angle  w. 

Draw  PiQi  and  P2Q2  parallel  to  OY,  to  meet  OX  in  Q^  and  Qg- 

Draw  P^R  parallel  to  OX  to  meet  P^Qi  in  i?. 
fY  VY 


Then         OQ,  =  x,,     OQ^^x^,     QiPi^Vi,     ^2^  =  2/2- 
.-.  P2R  =  ^2^1=  OQ,  -  0Q2=x,  -  x„ 
and  liP,=  QiPi-Q,R=QiPi-Q2P2  =  yi-y2' 

Also  Z  P2RP1  =  Z  OftPi  =  TT  -  (o. 


7]  DISTANCES  9 

From  the  triangle  P1RP2  we  have,  by  the  law  of  cosines, 

P,P^^  =  P^P^  +  RP,^  _  2P2R  .  RP^  cos  (tt  -  io). 
Whence  by  substitution,  since  cos  (tt  —  w)  =  —  cos  a>, 

PiP\  =  [(a^i  -  352)'^  +  (2/1  -  2/2)'^  +  2  (a5i  -  052) (Vi  - 1/2)  cos  «]2.  (1) 
When  the  axes  are  rectangular,  a>  =  90°  and  cos  w  =  0. 
Hence  for  the  distance  between  two   points  whose  rectangular 
coordinates  are  given,  we  have  the  very  useful  formula 

P^Pi  =  V(a?i- 072)2+ (2^1-2/2)2.*  (2) 

If  the  plus  sign  before  the  radicals  in  (1)  and  (2)  gives  P2P1,  the 
minus  sign  will  give  PiPi-  It  will  aid  the  memory  to  observe  that 
the  meaning  of  (2)  is  expressed  by  writing 

(Distancey  =  (EastingY  -\-  {Northingf. 

Cor.  If  P2  coincides  with  the  origin  X2  =  2/2  =  0,  and  equations 
(1)  and  (2)  give  for  the  distance  of  a  point  Pi(a7i,  2/1)  from  the  origin 

OJ*i  =  ^ici^  +  2/i2  +  2  xxvx  cos  CO,  for  oblique  axes,  (3) 


OP\  —  ViCi^  _|_  y^2^   for  rectangular  axes.  (4) 

EXAMPLES 

1.  Find  the  distance  between  (—  5,  3)  and  (7,  —  2). 

2.  Show  that  if  the  axes  are  inclined  at  an  angle  of  60°,  the  distance  between 
the  points  (-  3,  3)  and  (4,  -  2)  is  \/39. 

3.  Find  the  distance  from  the  origin  to  the  point  (—  2,  4)  when  the  axes  are 
inclined  at  angle  of  120°. 

4.  Find  the  lengths  of  the  sides  of  the  triangle  whose  vertices  are  (4,  1), 
(-2,  4),  and  (1,-2). 

6.  Show  that  the  four  points  (2,  4),  (1,  7),  (-  2,  4),  and  (-  1,  1)  are  the 
angular  points  of  a  parallelogram. 

6.  If  the  point  (a;,  y)  is  6  units  distant  from  the  point  (3,  4),  then  will 
JC2  +  y2  _  6  X  -  8  ?/  =  0. 

*The  student  should  convince  himself  of  the  generality  of  equations  (1)  and  (2)  by 
constructing  other  special  cases  in  which  the  given  points  lie  in  different  quadrants. 
He  will  thus  have  an  illustration  of  the  general  principle  that  formulsB  and  equations 
deduced  by  considering  points  lying  in  the  first  quadrant,  where  both  coordinates  are 
positive,  must,  from  the  nature  of  the  analytic  method,  hold  true  when  the  points  are 
situated  in  any  quadrant. 


10 


DISTANCES 


[8 


8.    The  distance  between  two  points  in  terms  of  their  polar  coordinates. 


Let  Piipx,  6y)  and  P2(p2>  ^2)  be  the  two  given  points. 
Then        OPr  =  p,,        OP,  =  p„        ZXOP,  =  e,,      ZXOP2  =  e2, 
and  Z  P2OP1  =  0,-  O^. 

Erom  the  triangle  PxOP^^  as  in  §  7,  we  have 

P^Pi  =  OPi'  +  OP2'  -2  0P,'  OP,  cos  P2OP1, 


.  ••  P1P2  =  ^Pi'^  +  P'2^  -  2  P1P2  cos  (61  -  62). 

Ex.  1.     Derive  equation  (2),  §  7,  from  equation  (1),  §  8. 
Expanding  the  last  term  and  squaring  (1),  §  8,  gives 

P1P22  =  pi2  +  p,^2  _  2(pi  cos  ^1)  (p2  cos  6-2)  -  2  (pi  sin  ^1)  (pa  sin  ^2). 
Substituting  the  values  given  in  equations  (1),  §  6,  we  have 
P1P22  =  Xi2  +  yi^  +  x^^  +  yo^  -2  xixa  -  2  yi^a- 


(1) 


.  •.     P1P2  =  V(xo  -  xi)2  +  (?/2  -  yxy\ 

Ex.  2.     Show  that  the  distance  between  the  points  (4,  90°)  and  (-3,  30°), 
is  V37. 

Ex.  3.     Find  the  distance  between  (2  a,  180°)  and  (-a,  45°).    . 

9.    To  find  the  coordinates  of  the  point  which  divides  the  line  join- 
ing two  given  points  in  a  given  ratio  (mi :  m<^. 

Let  Pi{x^,  2/1)  and  P^ix^,  2/2)  be  the  two  given  points,  and  let  P{x,  y) 
be  the  required  pointo 

Draw  PiQi,  PQ,  P2Q2  parallel  to  the  ?/-axis,  and  PR,  P^Ri  parallel 
to  the  ic-axis. 

Then  P^E^  =  x  —  x^,     PR  =  x.2  —  x, 

RiP=y-yx,     JKP2=2/2-2/. 


9] 


DISTANCES 


11 


From  the  similar  triangles  PiPIii  and  PP2R,  we  have 
PiP  _  PiRi  _  RiP  _'nh  _x  —  x^  _y  —  y^ 


PP.,       PR       RP,,      ma      x^-x      2/2 


y 


.-.  wii (x2  —  x)  =  m^ (x  —  Xi), 
and  mi  (2/2  -y)  =  m^  (y  -  2/1). 

Solving  (1)  and  (2)  for  x  and  y,  respectively,  we  obtain 


ac 


miX2  +  tn^ooi 


y  = 


miijQ  +  ^^22/1 


(1) 
(2) 

(3) 

TTxT'   ^-    i  +  x   •  W 

These   equations,  (3)   or  (4),  cover  all  cases,  the  division  being 

internal  or  external  according  as  A  is  positive  or  negative. 

It  P  be  the  middle  point  of  PiP,,  ni^  =  mo,  and  therefore  the 

coordinates  of  the  middle  of  a  line  joining  two  given  points  are 


If  we  let  A  =  mi :  mg,  equations  (3)  reduce  to  the  form 
«!  +  Xa?2  V\  +  X2/.2 


05  = 


»  =  |(a?i  +  £C2),    2/ =  2 (2/1 +  2/2)- 


(5) 


These  formulae,  (3),  (4),  (5),  are  independent  of  the  angle  between 
the  axes,  and  hold  for  both  rectangular  and  oblique  axes. 

Ex.  1.  Find  the  points  which  divide  the  line  joining  (2,  5)  and  (—6,  —  2) 
internally  in  the  ratio  8  :  4,  and  externally  in  the  ratio  2  :  9. 

Ex.  2.  In  what  ratio  is  the  line  joining  the  points  (2,  1)  and  (—  8,  6)  divided 
by  the  point  (  -  2,  3)  ?  by  the  point  (8,  -  2)  ? 


12 


AREAS 


£10 


Areas  of  Polygons 
10*    To  find  the  area  of  a  triangle  in  terms  of  the  coordinates  of  its 
vertices,  the  axes  being  inclined  at  an  angle  w. 
Case  I.    When  one  vertex  is  at  the  origin. 


Let  Pi(xi,yi),  Po(x2,y^  be  the  other  two  vertices.    Draw  PiQi,  P^Qo 
parallel  to  the  ^/-axis,  and  Q^R  perpendicular  to  ^9^2- 
Then         OQ^  =  x^,     0Q2  =  x,,     Q^P^  =  y^,     ^2^  =  2/2, 
RQ^  =  Q.2Q1  sin  0)  =  (x^  —  X2)  sin  w,  and 
A  OP1P2  =  A  OQ2P2  +  trap.  QsQi^i  A*  -  A  OQ^P^, 

=  i[0Q2  •  Q2P2  +  Q2QM2P2  +  QiA)  -  OQ, .  QiPJ  sin  CO, 
=  i  [a;22/2  +  (a^i  -  ^2)  (2/1  +  2/2)  -^i2/i]  sin  w, 


=  I  (Sx^iUi  -  ^iVi)  sin  (0 

in  the  notation  of  determinants. 


2/1 
2/2 


sin 


(1) 


*  The  area  of  the  trapezoid  ABCD,  in  which  the  non-parallel  sides  intersect,  is 

the  difference  of  the  areas  of  the  two  triangles  formed  by  the  diagonal  AC.    That  is, 

ABCD  =  ABC -ADC  =  ABE-  CDE. 

This  is  expressed  analytically  by  saying  that  the  area  is  the  algebraic  sum  of  the 

triangles.    The  base  CD  is  then  regarded  as  changing  its  direction  (and  sign)  with 

reference  to  AB  ;  for  in  going  along  the  sides  con- 
secutively in  the  order  ABCD  A,  the  base  CD  is 
traversed  in  the  same  direction  as  A  B,  which  is  not 
the  case  in  the  ordinary  trapezoid.  That  is,  when 
D  is  to  the  left  of  C,  both  the  base  CD  and  the 
area  of  the  triangle  ACD  are  positive,  say.  But  as 
D  moves  to  the  right,  both  CD  and  the  area  ACD 
>B    become  zero  and  change  sign  as  D  passes  through  C. 


10] 


AREAS 


13 


Case  II.    When  the  origin  is  not  a  vertex  of  the  given  triangle. 

Pa  \Y  P. 


Let  Pi(xi,  ?/i),  Po(x2,  2/2)?  ^3(^3)  y:i)  be  the  vertices  of  the  given 
triangle.     Draw  the  lines  OP^,  OP^,  OP^.     Then  by  Case  I  we  have 

^i,  yi 


A  OP1P2  =  i(a;i2/2  —  X2I/1)  sin  to=l 
A  OP.Ps  =  K^22/3  -  ^32/2)  sin  o>  =  i 
A  OPiPi  =  i(.T3.?/i  -  .T12/3)  sin  o)  =  ^ 


•'i'2j  2/2 
^3)  2/3 


Sin  o). 


sm  o). 


sin  o). 


.-.    A  PiP2^3=  i[(^*i2/2 -  a^iVi)  +  ('<^-22/3  -  ^3?/2)  +  (^^sZ/i  "  aTi^s)]  sin  o>     (2) 


=i 

( 

a^i,  2/1 

a^2, 2/2 

+ 

^'2,  2/2     , 

3^3,2/3 

a-*3,  2/3 

a^i,  2/1 

^1,  yi,  1 

=i 

^2, 2/2, 1 

sill  (U. 

^ 

h,  ys,  ^ 

sin  (u 


(3) 


When  the  axes  are  rectangular  sin  w  =  1,  and  equations  (1),  (2), 
(3),  respectively,  reduce  to 


A  O P1P2  =  I  (i»i?/2  -  OC27Jl)  =  I 


(4) 


A  P1P2P8  =  I  i^K^iVi  -  a^-22/i  +  ^'iV'i  -  «82/2  +  3582/1  -  ^iVfi)       (5) 


oci,  2/1,  1 

a?2,  2/2J  1 

=  k 

i»8,  2/8J  1 

Xi        .Tj,  2/1         3^2 

X2  —  x^,  y^     2/3 


(6) 


14  AREAS  [11 

11*  When  the  origin  is  within  the  given  triangle,  the  given 
triangle  includes  the  three  triangles  OP^P^,  OP^P^,  OP^P^  (§  10) ; 
hence  the  expressions  ^{x^^  —  ^22/i)>  ^(^'22/3  —  ^32/2)5  and  ^{x.^^  —  x^y^ 
must  have  the  same  sign.  When  the  origin  is  outside,  the  given 
triangle  does  not  include  all  of  these  triangles,  and  therefore  the 
above  expressions  can  not  have  the  sa^ne  sign. 

Suppose  a  person  to  start  from  0  and  walk  consecutively  around 
the  triangles  OP^P^,  OP2P3,  OP^P^  in  the  direction  indicated  by  this 
order  of  vertices.  This  imaginary  person  would  thus  walk  along 
each  side  of  the  given  triangle  once  in  the  same  direction  around  the 
figure,  as  indicated  by  P^P^P&y  and  along  each  of  the  lines  OP^,  OP2, 
OPsf  twice  in  opposite  directions.  When  the  origin  is  inside  the  given 
triangle,  he  would  walk  around  each  of  these  triangles  in  such  a 
manner  that  he  would  have  its  area  always  on  his  left  hand.  When 
the  origin  is  outside,  he  would  go  around  those  triangles  which  in- 
clude no  part  of  the  given  triangle,  in  such  a  manner  that  he  would 
have  their  area  ahcays  on  his  right  hand. 

Thus  direction  around  a  triangle  may  be  taken  to  indicate  the  sign 
of  its  area.     (See  footnote  under  §  10.) 

The  expressions  for  area  in  §  10  will  be  found  to  be  positive,  if 
the  vertices  are  numbered  so  that  in  passing  around  in  the  direction 
thus  indicated  the  area  is  always  on  the  left. 

Let  the  student  show  by  trial  that  (x^y^  —  x^jy^)  is  ±  according  as 
Z  P1OP2  is  ±;  Z  P1OP2  is  ±  according  as  the  cycle  OP^Po  is  ±. 

12.*  To  express  the  area  of  a  triangle  in  terms  of  the  polar  coordi- 
nates of  its  vertices. 

Let  Pi(pi,  ^1),  P2(p2)  ^2)?  ^sfe)  ^3)  be  the  three  vertices. 

Then  x^  =  pi  cos  61,     X2  =  p2  cos  62,     x^  =  p^  cos  9s, 

2/1  =  pi  sin  61,     2/2  =  p2  sin  O2,     2/3  =  Ps  sin  ^3.     [(1),  §  6.] 

Substituting  these  values  in  (5)  and  (6)  of  §  10  gives 
OP1P2  =  i  pip2  (sin  $2  cos  61  —  cos  62  sin  Oi)  =  |  pipa  sin  (62  —  ^1).  (1) 

PiP2Ps=  i  [piP2  sin  (82  -  e,)  +  pops  sin  (^3  -  O2)  +  PsPi  sin  (6,  -  6^)^     (2) 

From  (1)  it  follows  that  the  three  terms  of  (2)  represent,  re- 
spectively, the  areas  of  the  triangles    OP1P2,   OP2PS,  and   OP3P1. 


13] 


AREAS 


15 


The  signs  of  these  terms  are  the  eigns  of  the  angle  differences 
(since  p  can  always  be  made  positive),  and  we  therefore  have  an 
independent  proof  of  the  statements  in  §  11. 

Let  the  student  prove  (1)  and  (2)  directly  from  a  figure. 

13*    To  find  the  area  of  any  polygon  when  the  rectangular  coordi- 
nates of  its  vertices  are  known. 

LetPi(a;i,   2/1),    ^^2^   .^2),  -^sfe  Vs^  A  (^'4,   2/4)  ••• -^nC^n,   2/«)  be 
the  n  vertices  of  the  given  polygon.     Then,  we  have,  from  (5)  §  10, 


A  OP.Po 


A  OP,P,  =  i 


x^, 

2/1 

X2, 

2/2 

^3, 

2/3 

X,, 

2/4 

A  OP,P,  =  \ 
A  OP,P,  =  i. 


^2, 

?h 

•^3> 

2/3 

^4, 

2/4 

X3, 

2/5 

AreaPiPa     •  ^« 


A 

OP.P, 

=  \ 

a?„,  2/« 

^1,   2/1 

=H 

Xi,   2/1 

a^2,  2/2 

%  ^3 

a^3, 

^A1 

2/3 
2/4 

+ 

^A,     2/4 

+  ••• 

aJ„,  y„ 

a^5, 

^5 

aji. 

2/1 

(1) 


since  the  area  of  the  polygon  is  the  algebraic  sum  of  the  areas  of 
these  triangles.  This  formula  is  easy  to  remember,  but  by  expand- 
ing the  determinants  and  collecting  the  positive  and  negative  terms 
it  may  be  written, 

Area  PiPg  •••  1*„  =  \  [(0512/2  +  ^aVs  +  ^^V\  +  •••  ^nVi) 

-  (yii»2  +  2/2^53  +  Vti^i^  +  •••  yn«l)])  (2) 

which  gives  the  following  simple  rule  for  finding  the  area  of  a 
polygon  when  the  rectangular  coordinates  of  its  vertices  are  known : 

(1)  Number  the  vertices  consecutively,  keeping  the  area  on  the  left. 

(2)  Multiply  each  abscissa  by  the  next  ordinate. 

(3)  Multiply  each  ordinate  by  the  next  abscissa. 

(4)  From  the  sum  of  the  first  set  of  products  subtract  the  sum  of  the 
second  set  and  take  half  of  the  result. 

If  the  axes  are  oblique,  the  second  members  of  (1)  and  (2)  must 
be  multiplied  by  the  sine  of  the  angle  between  the  axes. 

The  law  of  the  sign  of  the  area  is  the  same  as  for  the  triangle. 


16  EXAMPLES   ON  CHAPTER  I  [13 

EXAMPLES  ON   CHAPTER  I 

Find  the  area  of  the  polygons  the  coordinates  of  whose  vertices  talcen  in 
order  are,  respectively, 

1.  (1,3),  (-2,  -4),  and  (3,  -1). 

2.  (2,  5),  (-6,  -2),  and  (-1,  5),  when  w  =  60°. 

3.  (4,  15°),  (-5,  45°),  and  (6,  75°). 

4.  (3,  -30°),   (-5,  150°),  and  (4,210°). 

5.  (2,  15°),  (6,  75°),  and  (5,  135°). 

6.  (-a,  ^tt),  (a,  I  it),  and  (-2a,  -|7r). 

7.  (a,  b  +  c),  (a,  b  —  c),  and  (—a,  c).  * 

8.  (a,  c  +  a),  («,  c),  and(— a,  c  — a). 

9.  (2,3),   (-1,4),  (-5,  -2),  and  (3,  -2). 

10.  (4,5),  (1,4),  (-2,6),  (-5,3),  (-2,-1),  (-3,-4),  (1,-2), 
(3,  -4),  and  (2,  1). 

11.  What  are  the  rectangular  coordinates  of  (4,  30°),  (—2,  135°), 
(-3,1^)? 

12.  What  are  the  polar  coordinates  of  (3,  -  4),  (-  5,  12),  (1,  3)  ? 

13.  Find  the  coordinates  of  the  points  which  trisect  the  line  joining  the 
points  (-2,  -1)  and  (3,  2). 

14.  Find  the  coordinates  of  the  point  which  divides  the  line  joining  (3,  —  2) 
and  (  —  5,  4)  internally  in  the  ratio  3  : 4. 

15.  Find  the  coordinates  of  the  point  which  divide:;  the  line  joining  (5,  3) 
and  (—  1,  4)  externally  in  the  ratio  3:2. 

16.  Find  the  length  of  the  sides  and  medians  of  the  triangle  (2,  6),  (7,  —  6), 
(—5,  —  1).     What  kind  of  a  triangle  is  it  ? 

17.  Find  the  length  of  the  sides  and  the  area  of  the  triangle  (3,  4),  (—1,  0), 
(2,  -  3).     What  kind  of  a  triangle  is  it  ? 

18.  Find  the  sides  and  area  of  the  quadrilateral  whose  vertices  taken  in 
order  are  (5,  -  1),  (-  1,  2),  (-  5,  0),  and  (1,  -  3).  What  kind  of  a  quad- 
rilateral is  it  ? 

Change  to  polar  coordinates  the  equations 

19.  x^  +  y^  =  r\  20 
21.    x^  =  y%2a-x).                               22, 
Transform  to  Cartesian  coordinates 
23.    d  =  tsin-^  m.                                      24. 
25.   p  =  a  sin  2  6.                                     26. 


y  =  xta,u  a. 

(x^^y^)(x-ay  = 

=  &2^2. 

p2  =  a2  sec  2  6. 

pi  =  ai  sin  |  d. 

13]  EXAMPLES  ON   CHAPTER  I  17 

Prove  analytically  the  following  theorems  : 
/ 

27.  The  diagonals  of  a  parallelogram  bisect  each  other. 

28.  The  lines  joining  the  middle  points  of  the  adjacent  sides  of  any  quadri- 
lateral form  a  parallelogram. 

29.  The  three  medians  of  a  triangle  meet  in  a  point,  which  is  one  of  their 
points  of  trisection. 

30.  The  lines  joining  the  middle  points  of  opposite  sides  of  any  quadrilateral 
and  the  line  joining  the  middle  points  of  its  diagonals  meet  in  a  point  and  bisect 
one  another. 

y 

31.  The  area  of  the  triangle  formed  by  joining  the  middle  points  of  the  sides 

of  a  given  triangle  is  equal  to  one-fourth  of  the  area  of  the  given  triangle. 

32.  If  in  any  triangle  a  median  be  drawn  from  the  vertex  to  the  base,  the 
sum  of  the  squares  of  the  other  two  sides  is  equal  to  twice  the  square  of  half 
the  base  plus  twice  the  square  of  the  median. 

33.  The  sum  of  the  squares  of  the  four  sides  of  any  quadrilateral  is  equal  to 
the  sum  of  the  squares  of  the  diagonals  plus  four  times  the  square  of  the  line 
joining  the  middle  points  of  the  diagonals. 

34.  Pi(xi,  yi),  Fiix^,  yz),  T-iixz,  2/3),  I'^ix^,  2/4)  ••  •  Tr,{x,,,  yn)  are  any  n 
points  in  a  plane.  PiPo  is  bisected  at  Qi  ;  QiP^  is  divided  at  Qz  in  the  ratio 
1:2;  ^2^4  is  divided  at  Q^  in  the  ratio  1:8;  ^jjPs  at  Q4  in  the  ratio  1 : 4,  and 
so  on  till  all  the  points  are  used.  Show  that  the  coordinates  of  the  final  point 
so  obtained  are 

a:i  +  a:2  +  a:3  -}-  a;4  +  ...  a:,.  ?/i  +  2/2  +  2/3  +  2/4  +  .  .  .  Vn 

and 


n  n 

Show  that  the  result  is  independent  of  the  order  in  which  the  points  are  taken. 
[This  point  is  called  the  Centre  of  3Iean  Position  of  the  n  given  points.] 


CHAPTER   II 


»  >o 


LOCI  AND  THEIR  EQUATIONS 

14.  It  has  been  shown  in  §  3  that  to  each  pair  of  values  of  x  and 
y  there  corresponds  in  all  the  plane  one  and  only  one  point,  and  that 
to  each  point  corresponds  one  and  only  one  pair  of  values.  Also,  if 
X  and  y  vary  independently  and  unconditionally  from  —  oo  to  oo, 
every  point  in  the  plane  will  be  obtained. 

If,  on  the  contrary,  one  or  both  of  the  coordinates  cannot  take  all 

values,  or  if  all  values  cannot  be 
independently  taken  by  both,  the 
point  cannot  move  to  all  positions 
in  the  plane. 

If,  for  example,  ic  >  0,  the  point 
X  (x,  y)  must  lie  to  the  light  of  the 
~  2^-axis ;  ii  x<0,  the  point  must  lie 
to  the  left  of  the  ^/-axis;  if  x  is 
neither  greater  nor  less  than  zero,  the 
point  can  lie  neither  to  the  right  nor 
to  the  left  of  the  ?/-axis ;  i.e.  if  x=0, 
the  point  must  lie  07i  the  2/-axis. 

15.  If  x>a,  the  point  (x,  y) 
must  lie  to  the  right  of  the  parallel 
AB,  which  is  a  units  to  the  right 
of  the  2/-axis ;  if  x<a,  the  point 
must  lie  to  the  left  of  AB.  There- 
fore, if  x  =  a,  the  point  will  lie  on 
the  line  AB. 

Ex.  1.  Where  will  the  point  {x,  y)  lie 
ifx>-3?    x<  -S?    x=  -S? 

Ex.  2.  Where  is  the  point  (x,  y)  if 
y>&?  ?/<6?  y  =  6?  y>-hf 
y<-b?     y=-b? 

18 


X  <0 


Y 

a 

A 

X  <Ca 

x:>a 
X 

0 

1 

B 

17] 


LOCI  AND   THEIR  EQUATIONS 


19 


16.  Draw  a  circle  with  centre 
at  the  origin  and  radius  equal  to  a. 

Then  the  point  P{Xf  y)  will  be 
outside,  inside,  or  on  this  circle 
according  as 

OP>a,    OP<a,   or    OP=a. 

But  OP'  =  a.-^  +  y'.     [(4),  §  7.] 

Therefore  the  point  P(x,  y)  is 
outside,  inside,  or  on  the  circle, 
according  as 

01^  -\-  y'  >  a%    x^  -\-y^  <  a-,     or     oc^  -\-  y^  =  a-. 

Ex.  1.  Write  down  the  conditions  that  the  point  (x,  y)  shall  be  outside, 
inside,  or  on  the  circle  whose  centre  is  at  the  origin  and  radius  3. 

Ex.  2.  What  are  the  conditions  that  the  point  (a;,  y)  shall  be  outside,  inside, 
or  on  a  circle  with  centre  at  (—  3,  1)  and  radius  4  ? 

Ex.  3.  Draw  a  circle  with  centre  at  (a,  6)  and  radius  r,  and  write  down  the 
conditions  that  the  point  (x,  y)  shall  be  outside,  inside,  or  on  this  circle. 

17.  Let  the  line  AOB  bisect  the  angle  XO  Y. 


Y 

X                       1 

Y 

B 

V    y 

P 

X 

< 

y 

/ 

y 

} 

/ 

y 

/ 

o 

./ 

/ 

Y' 

X' 


Then  every  point  on  AB  is  equidistant  from  the  axes.    Hence  the 
point  Pix^  y)  is  above  AB,  below  AB,  or  on  AB,  according  as 

y>x,    y<x,    or    y  = », 
or  according  as  y  —  x  >,  <,  or  =  0  j 

i.e.  according  as  y  —  a;  is  positive,  negative,  or  zero. 


20 


LOCI  AND   THEIR  EQUATIONS 


[19 


18.  Draw  CD  parallel  to  ABj  cutting  the  ^/-axis  in  E,  three  units 
above  0. 

Then  every  point  on  CD  is  three  units  farther  from  the  a>axis 
than  from  the  ?/-axis.  Therefore  the  point  P{x,  y)  will  be  above  CD, 
below  CD,  or  on  CD,  according  as 

?/>,  <,  or  =aj  +  3; 
i.e.  according  as  ?/  —  a;  —  3  is  positive,  negative,  or  zero. 


Y 

> 

'     p  y^ 

""    / 

q.  /" 

/ 

/ 

y 

/ 

E 

/ 

/ 

p 

y 

y 

^/         A. 

o 

Y' 

Ex.  1.  Draw  a  Hne  parallel  to  AB^  cutting  the  y-axis  two  units  below  0  ;  and 
write  down  the  conditions  that  the  point  (ic,  y)  shall  be  above,  below,  or  on 
this  line. 

Ex.  2.  What  are  the  conditions  that  the  point  (cc,  xj)  shall  be  above,  below, 
or  on  the  line  through  E  parallel  to  the  bisector  of  the  angle  X'OF? 

19.  Let  CD  be  the  perpendicular  bisector  of  the  line  joining 
A{-\,V)  and  i5(3,  -1). 

Then  all  points  on  CD  are  equidistant  from  A  and  B,  and  all 
other  points  are  not  equally  distant  from  A  and  B.  Hence  the 
point  P{x,  y)  will  lie  to  the  rujlii  of,  to  the  left  of,  or  on  CD, 
according  as  ^P  >,  <,  or  =  BP, 

or  according  as  AP"-  >,  <,  or  =  BP'^-, 

i.e.  according  as  [(2),  §  7] 

(a;  +  l)2  +  (2/-iy>,  <,or  =  (a:-3)2  +  (2/  +  iy;      " 
whence  2x  —  y  —  2>,  <,  or  =  0. 


20] 


LOCI  AND  THEIR  EQUATIONS 
Yl  p  /D 


21 


Ex.  1.  Find  the  conditions  that  the  point  (x,  y)  shall  be  above,  below,  or  on 
the  perpendicular  bisector  of  theiine  joining  (2,  3)  and  (—  1,  —  2). 

Ex.  2.  What  is  the  condition  that  (x,  y)  shall  be  on  the  perpendicular 
bisector  of  the  line  joining  (a,  6)  and  (c,  d)  ? 

\^^      20.   The  examples  in  §§  14-19  illustrate  certain  general  principles, 
of  which  we  will  here  make  only  a  preliminary  statement. 

I.  All  points  whose  coordinates  satisfy  an  equation  of  condition 
(not  an  identity)  lie  on  a  certain  line ;  and  conversely,  if  a  point  lies 
on  a  fixed  line,  its  coordinates  must  satisfy  an  equation. 

II.  Points  whose  coordinates  satisfy  a  condition  of  inequality  do 
not  lie  on  any  fixed  line. 

If /(a;,  //)  be  used  to  represent  any  expression  containing  the  two 
variables  x  and  y  and  certain  constants,  these  principles  may  be 
stated  more  definitely,  as  follows : 

I.  All  points  whose  coordinates  make /(a;,  y)  =  0,  lie  on  a  certain 
line ;  and  conversely,  the  coordinates  of  all  points  on  this  line  make 

/(^,  y)  =  0. 

II.  If  f(oci,  ?/i)  >  0  and  /(ajo,  2/2)  <  ^j  t^©  *wo  points  (ar„  y,)  and 
(x2,  y>^  lie  on  opposite  sides  of  the  line  the  coordinates  of  whose 
points  make  /(a;,  y)  =  0. 

Hence  every  line,  as  well  as  the  axes  of  coordinates,  is  said  to 
have  a  positive  and  a  negative  side. 


22  LOCI  AND   THEIR   EQUATIONS  [20 

Def.  The  locus  of  a  variable  point  subject  to  a  given  condition  is 
the  place,  i.e.  the  totality  of  positions,  where  the  point  may  lie  and  sat- 
isfy the  given  condition. 

Def.      The  line  (or  lines)  containing  all  points,  and  no  others,  whose 
coordinates  satisfy  a  given  equation  is  called  the  Locus  of  the  Equation ; 
conversely,  the  equation  satisfied  by  the  coordinates  of  all  points  on  a 
certain  line  (or  lines)  is  called  the  Equation  of  the  Line,  or  the  Equation  y^ 
of  the  Locus. 

Def.  That  part  of  the  plane  containing  all  points,  and  no  others, 
whose  coordinates  satisfy  a  given  inequation  is  the  Locus  of  the 
Inequatio7i. 

Thus  the  Locus  of  a  point  in  Plane  Geometry  is  not  ahvays  a 
line. 

In  the  examples  of  §§  14-19  only  Cartesian  coordinates  have  been 
used,  but  the  fundamental  principles  there  illustrated,  and  also  the 
above  definitions,  hold  for  all  systems  of  coordinates. 

Let  the  student  give  some  similar  illustrations  with  polar  co- 
ordinates. 

EXAMPLES 

What  is  the  locus  of 

1.  x2  +  ?/2  =0  ?    x'-^  +  2/2  >  0  ?    x^  -\-y'^<0? 

2.  X=  VX^  +  ?/2  ?      X  >   Vx^  +  ?/2  ?      X  <  \/x2  -f-  ^2  ?  i 

3.  p  =  a  sec  ^  ?    p  >  a  sec  ^ ?    p  <  a  sec  ^ ?  il  )-  ■ 

4.  p  =  &  CSC  0  ?    P  >  &  CSC  ^  ?    p  <  6  CSC  ^  ?       ' 


J- 

5.   4  <  x2  +  ?/  <  9  ?  ^i^^f^  --^  ' 


6.  9<(x-2)2+(?/_3)2'<16?/.r^P>' 

7.  a  sec  ^  <  p  <  6  sec  ^  ?  i-j  //^ 

8.  p  =  a  COS  ^  ?    p  >  a  cos  ^ ?  X < a^fffid ?        fr^^ * 

9.  acos0<p<  6cos^?  ^"jMj' 

10.  p  =a  sin  ^  ?    p  >  a  sin  ^  ?  ^^(z  sin J^m'*^ — 

11.  P  =  a?    p>a?    p<a?^X^>/£__ 

12.  What  is  the  locus  of  a  poiWmpving  so  that  the  sum  of  its  distances  from 
the  Unes  x  =  0  and  x  =  3  is  1,  2,  3,  4  ? 


21]  LOCI   AND   THEIR  EQUATIONS  23 

To  Find  the  Locus  of  a  Given  Equation 

21.  If  the  locus  of  an  equation  is  a  straight  line,  the  locus  is 
easily  drawn;  it  is  only  necessary  to  locate  two  points  on  it 
(preferably  the  intersections*  with  the  axes)  and  draw  a  straight 
line  through  these  points.  Likewise,  if  the  locus-is  a  circle,  the 
complete  locus  can  be  drawn  when  the  centre  and  radius  are  known. 

It  will  be  shown  farther  on  that  straight  lines  and  circles  can 
easily  be  recognized  by  the  forms  of  the  equations. 

In  general,  having  given  an  equation  of  condition  between  the 
coordinates  (in  any  system)  of  a  variable  point,  we  may  assign  any 
value  we  please  to  one  coordinate  and  find  a  corresponding!  value,  or 
values,  of  the  other.  To  every  such  pair  of  corresponding  value's  will 
correspond  a  definite  point  of  the  locus.  Since  these  pairs  of  values 
may  be  as  numerous  as  we  please,  we  can  in  this  way  locate  as  many 
points  of  the  locus  as  we  please.  A  smooth  curve  drawn  through 
these  points  will  be  an  approximation  to  the  locus  of  the  given  equa- 
tion. The  degree  of  approximation  will  depend  upon  the  proximity 
of  the  points  thus  located.  This  method  of  constructing  a  locus  is 
applicable  to  any  equation  that  can  be  solved  for  one  of  the  variables, 
and  is  called  Plotting  $  an  Equation,  or  Plotting  the  Locus  of  an 
Equation.     The  steps  of  this  process  are  as  follows : 

*  Unless  both  intersections  are  near  the  origin,  when  the  line  will  be  inaccurately 
determined,  or  both  at  the  origin,  when  its  direction  will  be  quite  undetermined. 

t  "  Corresponding  values  "  of  the  variables,  x  and  y  say,  involved  in  a  given  equa- 
tion are  a  pair  of  values  of  x  and  y  which  satisfy  the  equation. 

X  The  logic  of  the  process  of 
plotting  is  that  of  induction,  and 

should  be  so  recognized  by  the  ,'       ;  •'*.,      \  .'' 

student.  Given  the  points  A,  B, 
C,  D,  E,  F  on  n  curve;  then,  in 
the  absence  of  further  knowledge, 
we  take  as  a  probable  approxi- 
mation a  smooth  curve  drawn 
through  them  like  the  full  curve 
in  the  figure.    We  are  not  war-  ' '  ' 

ranted  in  drawing  such  a  curve  as  the  dotted  one  through  the  points,  because  it  is 
unlikely  that,  taking  points  at  random  on  such  an  irregular  curve,  tlu-  jxisititui  of 
these  points  should  fail  to  disclose  any  of  the  irregularity.  The  student  should  also 
be  warned  that  sudden  changes  of  slope  or  curvature  areas  unlikely  as  sudden  changes 
in  the  value  of  an  ordinate. 


24 


LOCI  AND  THEIR  EQUATIONS 


[22 


\ 

T 

/ 

> 

L 

/ 

\ 

/ 

L 

P) 

V 

/ 

\ 

\ 

/ 

p. 

X 

> 

X 

O 

y 

p. 

p. 

r. ' 

X  = 

^8 

-6 

-4 

— 

2 

y  = 

6.8 

3.4 

.8 

-  1 

X  = 

2 

4 

6 

8 

y  = 

-2.2 

-1.6 

.2 

2 

(1)  Solve  the  equation  with  respect  to  one  of  the  coordinates. 

(2)  Assign  to  the  other  coordinate  a  series  of  values  differing  but 
little  from  each  other. 

(3)  Find  each  corresponding  value,  or  values,  of  the  Jirst  coor- 
dinate. 

(4)  Locate  the  point  corresponding  to  each  pair  of  corresponding 
values  thus  found. 

(5)  Join  these  points  in  order  by  a  smooth  curve,  and  this  curve 
will  be  approximately  the  required  locus.  If  there  be  doubt  how  to 
fill  up  any  of  the  intervening  spaces,  interpolate  more  points. 

22.   Illustrative  Examples. 

Ex.  1.    Plot  the  locus  of  the  equation  lOy  =  a;^  —  3x  —  20. 
Assigning  to  x  values  from  —  8  to  +  10,  differing  by  two  units,  we  iind  the 
following  pairs  of  values  of  x  and  y  to  satisfy  the  equation : 

0 
-2 
10 
5 
Plotting  the  corresponding  points 
Pi,  Pa,  Pa,  etc.,  and  drawing  a  smooth 
curve  through  them  in  the  order  of  the 
increasing  values  of  x,  we  find  the  locus 
to  be  approximately  the  curve  drawn 
in  the  figure. 

Ex.  2.  Plot  the  locus  of  the  equation  y'^=\.x. 

Solving  for  y  gives  y  =  ±  2  ^  x. 

When  X  =  0,  1,  4,  9,  ...  to  od, 

y  =  0,  ±2,   ±4,   ±6  .  .  .  to  ±   00, 

The  corresponding  points  of  the  locus  are 
0(0,  0),  Pi(l,  -2),  P2(l,  2),  P3(4,  -4), 
P4(4,  4),  PcCO,  -  6),  and  PeCO,  6).  .  .  . 

When  X  is  negative,  y  is  imaginary.  There- 
fore no  points  of  the  locus  lie  to  the  left  of  the 
?/-axis.  For  every  positive  value  of  x  there 
are  two  values  of  y  numerically  equal  but 
opposite  in  sign.  Hence  the  two  correspond- 
ing points  of  the  locus  are  equidistant  from 
the  X-axis.  As  x  increases,  both  values  of  y 
increase  numerically. 


Y  ^    P» 

jpj/. 

[ x_ 

o 

_ 

^***«.  ^  p." 


22] 


LOCI  AND   THEIR   EQUATIONS 


25 


Therefore  the  locus  cannot  be  such  a  curve  as  that  represented  by  the  dotted 
line,  but  must  be  approximately  that  indicated  by  the  full  line. 

Ex.  3.   Plot  the  locus  of  the  equation  25(x  -  1)2  +  \Q{y  -  3)2  =  400. 

Solving  for  y  gives  y  =  3  ±  fVlG  —  (x-l)2. 

This  form  of  the  equation  shows  that  y  is  imaginary  when  a;  <  —3,  or  a:  >  5, 
since  16  —  (oj—  1)2  is  then  negative  ;  and  when  x  is  neither  less  than  —  3  nor 
greater  than  5  there  are  two  real  unequal  values  of  y,  one  found  by  using  the  + 
sign  before  the  radical,  the  other  by  using 
the  —  sign.  Hence  the  locus  lies  between 
the  two  parallel  lines  a;  =  —  3  and  x  =  5. 

The  equation  is  satisfied  by  the  follow- 
ing pairs  of  values  of  x  and  y : 

-3       -2  -1  0 

3  6.3  7.3  7.8 

3       -    .3        -1.3      -1.8 


X 

y 
y 

X  = 

y  = 
y  = 


2 

7.8 

1.8 


3 

7.3 
1.3 


4 

6.3 
-  .3 


7                    V 

l--\ 

P 

The  corresponding  points  are  P(—  3,  3), 
Pi(-2,  6.3),  P2(-2,  -  .3),  etc.,  and  the 
locus  Is  the  curve  shown  in  the  figure. 

Ex.  4.   Plot  the  locus  of  the  equation,  p  =  2a  sin  d. 

Here  p  has  its  greatest  value  when  sin  d 
has  its  greatest  value,  i.e.  when  d  =  \ir. 
As  d  increases  from  0  to  J  tt,  sin  ^  in- 
creases from  0  to  1,  and  p  increases  from 
0  to  2a  ;  as  ^  increases  from  ^tt  to  tt,  sin  0 
decreases  from  1  to  0,  and  p  decreases 
from  2a  to  0.  Hence  the  locus  starts 
from  the  origin  and  returns  to  the  origin 
as  6  is  made  to  vary  from  0  to  t. 

Assigning  to  6  values  from  0  to  180°, 
differing  by  30*^  we  find  the  following 
points  are  on  the  locus  : 

0(0,   0),   A{a,    30°),    P(aV3,   60°), 
C(2a,  90°),  D{a^y  120^"^),  E{a,    150°), 
and  0(0,  180°). 
The  complete  locus  is  the  curve  shown  in  the  figure. 

Ex.  a.  Show  that  the  points  A,  B,  .  .  .  all  lie  on  a  circle  tangent  to  OX  at 
O  and  whose  radius  is  a.  Show  also  that  every  point  on  this  circle  satisfies  the 
given  equation. 


26 


LOCI   AND   THEIR   EQUATIONS 


[22 


Ex.  6.  Show  that  the  same  circle  will  be  described  as  6  varies  from  180°  to 
360°  ;  also  as  6  varies  from  any  value  a  to  a  -h  v. 

We  have  in  this  example  an  illustration  of  a  characteristic  property  of  equa- 
tions in  polar  coordinates  containing  a  periodic  function  of  6.  In  such  equations 
p  takes  all  possible  values  as  6  varies  through  a  limited  range  of  values  called  the 
period  of  the  function.  The  complete  locus  is  described  at  least  once  as  6  varies 
through  this  period,  and  is  repeated  as  6  varies  through  any  other  equal  period. 

The  period  of  sin  0  is  2  tt  ;  hence  p  takes  all  possible  values  from  —2a 
to  +  2  a  as  ^  varies  from  0  to  2  tt.  The  whole  circle  is  described  tivice  as  6 
varies  through  this  period,  once  as  6  varies  from  0  to  tt  with  p  positive,  and  once 
as  6  varies  from  tt  to  2  tt  with  p  negative.  Also  the  whole  circle  is  described 
twice  if  6  starts  from  any  value  and  varies  through  2  tt  in  either  direction. 
Ex.  5.  Plot  the  locus  of  the  equation  p  =  sin  2  6. 
This  equation  I&  satisfied  by  the  following  pairs  of  values  of  p  and  6 : 

e  =  45°,  225°,     p=l. 
e  =  135°,  315°,     p=  -\. 
e  =  30°,  60°,  210°,  240°, 
P  =  ly/S. 

6  =  120°,  150°,  300°,  330°, 
P  =  -  I  V3. 
e  =  15°,  75°,  195°,  255°, 


e  =  105°,  165°,  285°,  345°, 

P=-\- 

e  =  0°,  90°,  180°,  270°,  360°, 

P  =  0. 

The  corresponding  points  are 
found  by  drawing  three  circles 
with  centres  at  O  and  radii  i,  |  V'^,  and  1,  and  then  drawing  radii  dividing  these 
circles  into  arcs  of  15°.     The  locus  is  the  four-leaf  curve  shown  in  the  figure. 

As  e  varies  from  0  to  2  tt,  the  four  leaves  are  described  in  the  order  1,  2,  3,  4, 
and  in  the  direction  indicated  by  the  arrow  heads. 

EXAMPLES 

Plot  the  loci  of  the  following  equations  :  * 

(2x-Sij  -    6  =  0.  W 
/       1.   \4x-6y-    6  =  0.\  2. 

[Qx-dy +  27  =0. 


2x-\-Sy  +  5  =  0. 
3x-2?/-12=0. 
5x4-2?/-    4=0.  J 


*  For  convenience  in  plotting  loci  the  student  should  be  supplied  with  "  coordinate 
paper,"  both  "  rectangular"  and  "polar." 

t  Loci  grouped  under  the  same  number  should  be  plotted  on  the  same  diagram. 


22] 


LOCI  AND   THEIR  EQUATIONS 


27 


r 


r  2  a;  +  9  y  +  13  =  0. 

3. 

y  =  lx-^. 
2y-x  =  2. 

7. 

6  a;2  +  5  a;?/  -  6 1/2  =  0. 

8. 

[x2-2/2  =  4.  J 

'  a;?/  =  2. 
x?/  =  -  2. 

11. 


r4(x+i)=i 


2)^ 


10y=(x+l)2. 


4.  (a;-4)(y  +  3)=0. 
6.  (x2-4)(y-2)  =0. 
6.  x2- 2/2  =  0.    4x2-«/2_o. 
f  a:2  4.  y2  _  25. 

10.       (a;-8)2+(?/-4)2  =  25. 

I  (a; -4)-' +(2,-2)2  =  5. 
12.  «/  =  x3-4x2-4x+16. 


MS.  ( !=(,^:-!r- 


14. 


[2/2=(x2-4)2.   J 


y  = 


a:4  _  20  a;2  +  64. 1 
x4  -  20  a;2  +  64.  J 


^5.  (x2  +  y2)2  =  a2(x2-y2). 


16.  //  =  x,   x2,   x3,   x*,    x^,  ...  x«.    x  =  2/,   2^2^    2/8,   2/*,   y^,--   y". 

Note  the  effect  of  interchanging  x  and  y  ;  e.gr.  the  locus  of  x  =  ?/*  is  obtained 
from  the  locus  of  y  =  x^  by  revolving  the  plane  through  180°  around  the  line 

17.  y  =  (x-l),  (x-l)2,  (x-l)3.      y.  18.  2/  =  x3,  x3-x,  x8  +  x.      -  f 

/"  19.  2/^  =  X,  x2,  x^,  x*.  20.  2/  =  sin  x,  cos  x,  sin-i  x,  cos"*  x. 

21.  2/  =  tan  X,  cot  x,  tan-i  x,  cot-i  x.         ^2.  2/  =  sec  x,  esc  x,  sec-^  x,  csc-i  x. 


-f  •  23.  2/=sin  2  x,  sin  ^- ,  |  sin  2  x,  2  sin  ^ .    24.  2/  =  6  sin  -,  6  sin 


x  +  c 


25.  p  =  sin  ^,  cos  0,  sec  0,  esc  ^. 

27.  p  =  cos  2  e,  cos  3  0,  cos  4  ^. 

/  29.  p  =  sin  \  e,  cos  ^  6. 

^31.  p=:acos^  +  ?). 

^^33.  2/  =  2^  log2X. 

fZb.  y  =  a^,  logaX.    (a>,  =,  <1.) 


2  a  a 

f  26.  p  =  sin3^,  8in4tf. 


28.  p  =  tan  ^,  cot  d. 
30.  p=        ^ 


6 


1  —  cos  d    3  —  2  cos  ^ 
32.  p2  =  sec  2  ^,  CSC  2  ^.     (Cf.  No.  9.) 
34.  2/  =  10^,  logio  X. 
i  36.  2/  =  2*,  2-^  i(2'  +  2-'). 


37.  2/  =  e«,  e  «,  ^  (e«  +  e  «).     Catenary,  if  e  =  2.7  +. 


38.2/  =  ^^,  (x-l)(x-2), 
x-3  x-3 

X    40.  y  =  ^+2,(^-l)(^-3). 
'^  ^      x  +  3  x-2 

42    y^(a;-l)(x.-3)(x-5) 
(x-2)(x-4)(x-6) 

44.y-(a;-l)(a;-3)(x-5) 
(x-2)(x-4) 


89    2/=(^-^)(^-^),    (^-^)(»-3). 
•  ^      (x-3)(x-4)      (x-2)(x-4) 

J    ,^(x+l)(x-2)      (x  +  2)(x-4) 
(x  +  3)(x-4)     (x-l)(x-3) 

48    ^^(a;-H)(x-4)(x-6) 
(x-l)(x  +  2)(x-3) 

46.  (x-l)(x  +  .3)(x-6), 

(x-2)(x-4) 


28  LOCI  AND   THEIR  EQUATIONS  [23 

^  46.    ,  =  ^^.  47.    ,  =  — ^ -e  48.    y=j^,^y 

«•  y = (^^rry/      ^«-  ^  =     (x--2)2    •      5^-  ^  =  (x-2)(x-3y 

'  52.   Are  the  points  (3,  60°)  (f ,  —  90°)  on  the  same  or  opposite  sides  of  the 

loci  of  Ex.  30  ?  ,     ^ 

53.    Which  of  -the  following  loci  pass  throug^the  origin  ?^yuiu&JO     ^-^^^^^"^^ 
^        4.  (1 )  2  X  +  3  y  =  O.i-  (4)  2/2  -  a2a;2>^5?^7)  yi  =  ^a^\\ 
0  Lv<^^)  iK^  +  2/2  =  1.       (5)  ax  +  &?/  +  c  =  0.   (8)  ^2  ^  4a  (x  +  a).)^ 
iT       f_(3)  2/  =  3x2-x.      (6)  ax2  +  &?/2  =  1.      (9)  (x  -  a)2  +  (y  -  6)2  =  a2  +  62. 

.  kv'^^^^^  What  is  the  necessary  and  sufficient  condition  that  the  locus  of  an  equation 
N\        in  Cartesian  coordinates  shall  pass  through  the  origin  ? 

The  Use  of  Graphic  Methods 

23.  It  has  been  shown  in  §§  14-20  that  whenever  the  relation 
between  two  quantities,  whose  values  depend  upon  one  another,  can 
be  definitely  expressed  by  an  equation,  then  the  geometric  or  graphic 
representation  of  this  relation  is  given  by  means  of  a  curve.  Such 
a  curve  often  gives  at  a  glance  information  which  otherwise  could 
be  obtained  only  by  considerable  computation ;  and  in  many  cases 
reveals  facts  of  peculiar  interest  and  importance  which  might  other- 
wise escape  notice. 

The  use  of  graphic  methods  in  the  study  of  physics,  analytical 
mechanics,  engineering,  and  many  other  branches  of  scientific  inves- 
tigation, is  already  extensive  and  is  rapidly  increasing.  Graphic 
methods  can  be  used,  however,  not  only  in  examples  where  the 
equation  connecting  the  two  variable  quantities  is  known,  such  as 
those  already  given,  but  also  in  examples  where  no  such  relation  can 
be  found ;  in  these  latter  cases  the  graphic  method  furnishes  almost 
the  only  practical  means  of  studying  the  relations  involved. 

Comparative  statistics,  and  results  of  experiments  and  direct 
observations,  can  frequently  be  more  concisely  and  forcibly  repre- 
sented graphically  than  by  tabulating  numerical  values.  The  fol- 
lowing are  simple  examples  of  this  kind: 

1.  The  following  table  shows  the  net  gold  (to  the  nearest  million  of  dollars) 
in  the  U.  S.  Treasury  at  intervals  of  one  month,  from  Jan.  10,  1893,  to  Oct.  31, 
1894  (Report  of  the  Sec.  of  the  Treas.,  1894,  p.  8) : 


>3J 


LOCI  AND   THEIR  EQUATIONS 


29 


1898 

Millions 
of  Dollars. 

1898 

Millions 
of  Dollars. 

1894 

liiUions 
of  Dollars. 

1894 

MiUiuus 
of  Dollars. 

Jan.   10 
Feb.   10 
Mar.  10 
Apr.  10 
May   10 
June  10 

120 
112 
102 
106 
99 
91 

July  10 
Aug.  10 
Sept.    9 
Oct.   10 
Nov.  10 
Dec.    9 

97 
104 

98 
87 
85 

84 

Jan.   10 
Feb.   10 
Mar.  10 
Apr.  10 
May   10 
June    9 

74 

104 

107 

106 

92 

69 

July   10 
Aug.  10 
Sept.  10 
Oct.    10 
Oct.    81 

65 

62i 

56 

60 

61 

Using  time  (in  months)  as  abscissas,  and  dollars  (1,000,000  per  unit)  as 
ordinates,  the  separate  points  represented  by  the  table  have  been  plotted 
(Fig.  1)  and  then  joined  by  a  smooth  curve. 


100 


50 


■^ 


;5 


123456789    10    11 


2     3     4     5     6     7 

1894 


8     9    10    11    12 


Fig.  1. 
In  this  example  the  curve  gives  no  new  information,  but  it  presents  in  a  much 
more  concise  form  the  information  given  by  the  tabulated  numbers.     Observe 
also  that  if  the  points  are  inaccurately  located,  the  diagram  becomes  not  only 
worthless,  but  misleading. 

2.  An  excellent  example  of  the  use  and  advantages  of  the  graphic  method 
of  representing  comparative  statistics  is  found  in  the  large  engraved  plate  placed 
under  the  front  cover  of  the  Annual  Report  of  the  Secretary  of  the  Treasury 
for  1894.  This  plate  presents  on  a  single  sheet  information  that  covers  several 
pages  when  expressed  in  tabulated  numbers.  All  of  the  curves  given  on  this 
plate,  except  one,  are  shown  (on  a  smaller  scale)  in  Fig.  2.  This  figure  should 
be  carefully  studied,  and  if  possible  the  original  plate  should  be  consulted. 

3.  The  curves  in  figures  1  and  2  were  constructed  by  locating  separate  points 
and  then  drawing  a  smooth  curve  through  these  points.  Such  curves  give  no 
new  information,  but  represent  graphically  information  already  ascertained. 

In  some  cases,  however,  curves  can  be  drawn  mechanically.  When  this  is 
possible  the  curve  is  constructed,  not  for  the  purpose  of  exhibiting  facts 
previously  known,  but  for  the  purpose  of  obtaining  new  information.  For 
instance,  in  the  stations  of  the  U.  S.  Weather  Bureau  an  instrument  called 


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..^      c    ^ 

J           L    w 

t    -       ^;^      - 

t           I    2 

_^L 52>   _-- 

7^ 1  •...: 

\            , 

II     ":^              ""s.     *• 

i           i_    § 

-  _   _  ^_      -  ^\-i"'--i  " 

j               '^    1         00 

•^         \     '... 

/[                 1 

^  _  _  _  ^ — ^_ 

VT" 

III      I        I    V   I        It          I^^ 

V             "~     00 

r         ^           ^^ 

f                              ^           r-l 

_    ^^u.      4:i\    :±  • 

-  -    _      _       --.      -I  _ 

3     i 

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'                    ^2 

_^ 

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J                     1 

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\^~       la 

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_      _    \     _._    ^:    2 

1 

^    _  __    1^  _^ 

1 

\ 

•                                      -l       00 

Z'          ^^ 

1                     '    2 

_             /-             i 

;                                     1       '^ 

-  \ 

«                                    1 

^            ^ 

"^'^      SIS 

^      ^        I      1    2 

>.                           *» 

_:i  :l-  l_  i 

3                                                  -^    - 

i    •.    L    j  ^ 

2  i                         i^r 

\        .'7        1     CO 

c     "3     1      si    ;S         ^__ . 

"Vt       T        «     "^ 

Sk       M       .»j        <«        3        C                                                 1*-. 

y     ^     1 

1    ^    "    1     2     5        -              "2 

t  *  »■  5"    '   s 

S    2    •-    G    H    1        -              -     / 

V   -i-i-    '    ^ 

h    ^     2     c     c    °                             ^- 

=  >  3  ■;;  "c  -                    **'^ 

jC      !!•     ' 

^    w    i     «    -2    Z                                              J/ 

V       \     "^     m 

^I           ^.   11    2 

O^UQCQg                                                  ,*"" 

N  U 

2    1     ?     ^     ^     2        I              "^"' 

-1     ^^     «s 

1     s  ^ 

SlE  2 

-      <N      rri      ^      in      N«5                                                           V 

lE 

\«B    00 

-^a  - 

•-H  eS    O 


23] 


LOCI   AND   THEIR   EQUATIONS 


31 


the  Thermograph*  constructs  automatically  a  curve  which  shows  the  continuous 
variation  of  the  local  temperature.  Similarly  the  Barograph*  records  the  varia- 
tion of  the  barometric  pressure,  etc. 

10  11 


Fig.  3.  — Thermographs  for  Aug.  9-10  and  Sept.  27-28,  1899,  at  Lincoln,  Neb. 
Mon.  13  Tu.  Wed.  Th.  Fri.  IT 


XII 


Mt 


XII 


Mt        XII 


Mt 


XII 


Mt 


XII 


Mt 


28 


-4-t. 


(IllililllllllllllllllllllilM 


Fig.  4.  — Barograph  Sheet,  March  13-17,  1899.  at  Lincoln,  Neb. 

Figures  3  and  4  are  copies  of  curves  thus  constructed  in  the  local  station  at 
Lincoln,  Neb.  The  upper  curve  in  Fig.  3  sliows  the  temperature  from  10  p.m. 
Aug.  8,  1899,  to  9  a.m.  Aug.  11,  1899 ;  the  lower  from  11  p.m.  Sept.  26,  1899,  to 
8  A.M.  Sept.  29,  1899.  Interpret  these  curves.  Notice  especially  the  record 
from  6  P.M.  to  midnight  Aug.  10. 

The  varying  pressure  on  the  piston  in  the  cylinder  of  a  steam  engine  is  deter- 
mined in  the  same  way  by  means  of  a  similar  instrument,  called  an  Indicator.* 

4.  Exhibit  graphically  the  information  contained  in  the  following  table  of 
wind  velocities  for  Jan.  20  and  June  15  and  25,  1894  : 

♦For  a  description  and  cut  of  the  "  Thermograph,"  "  Barograph,"  and  '*  Indicator," 
see  these  words  in  the  Century.  Standard,  or  Webster^s  International  Dictionary. 


32 


LOCI  AND  THEIR  EQUATIONS 


[24 


Day 

12-1 

1-2 

2-3 

3-4 

4-5 

5-6 

6-7 

7-8 

8-9 

9-10 

10-11 

11-12 

Jan.  20,  A.M.  .  . 
June  15,  A.M.  .  . 
June 25,  a.m.  .  . 

8 
15 
17 

6 
11 
14 

7 

8 

13 

7 
10 
13 

8 

8 

11 

9 
9 
23 

12 

3 

23 

15 
3 
13 

15 
11 
9 

19 
15 
4 

12 

17 

2 

21 
21 

10 

Jan.   20,  P.M.  .  . 
June  15,  P.M.  .  . 
June  25,  p.m.  .  . 

22 
15 
12 

22 
21 

15 

18 
22 
11 

19 
20 
12 

14 
17 
12 

9 
17 
5 

6 
12 

1 

7 
5 
3 

5 
5 
6 

6 
6 

7 

5 
6 

7 

4 
3 
3 

Intersection  op  Loci 

24.  To  find  the  points  of  intersection  of  two  loci  when  their  equations 
are  knoivn. 

Since  the  points  of  intersection  of  two  loci  lie  on  both  curves, 
their  coordinates  must  satisfy  both  equations.  Therefore,  to  find 
the  coordinates  of  the  points  of  intersection  of  two  loci  we  treat 
their  equations  simultaneously,  regarding  the  coordinates  as  the 
unknown  quantities,  and  thus  find  the  values  of  the  coordinates 
which  satisfy  both  equations.  A  pair  of  values  which  satisfy  both 
equations  are  the  coordinates  of  a  point  of  intersection  of  the  two 
loci. 

If  the  equations  are  both  of  the  first  degree,  there  will  be  but  one 
pair  of  values  of  coordinates  satisfying  them,  and  therefore  but  one 
point  of  intersection  of  the  loci. 

If  one  or  both  of  the  equations  be  of  a  higher  degree  than  the 
first,  there  will  be  several  pairs  of  roots,  and  one  point  of  intersec- 
tion for  ea^h  pair.  The  loci  will  then  have  several  points  of  inter- 
section. 

If  of  a  pair  of  roots  even  one  is  imaginary,  there  is  no  correspond- 
ing real  point  common  to  the  two  loci.  We  then  say  the  intersection 
is  imaginary. 

Since  imaginary  roots  of  equations  always  occur  in  pairs,  imagi- 
nary intersections  of  loci  always  occur  in  pairs,  and  hence  the  passage 
from  a  real  pair  of  intersections  to  an  imaginary  one  is  through  a 
coincident  pair.  Suppose,  for  example,  that  a  straight  line  intersects 
a  circle  in  two  real  points.  If  the  line  be  moved  so  that  it  becomes 
tangent  to  the  circle,  the  two  points  of  intersection  coincide  in  the 
point  of  contact.  If  the  line  be  moved  still  farther,  the  intersections 
are  said  to  become  imaginary. 


26]  LOCI  AND   THEIR  EQUATIONS  88 

25.  Intercepts  on  the  axes  of  coordinates. 

This  is  a  special  and  very  important  case  of  the  preceding  section 
in  which  one  of  the  given  equations  is  a;  =  0,  or  2/  =  0. 

To  find  the  points  of  intersection  of  a  curve  with  the  a>axis,  put 
y  =  0  in  the  equation  of  the  curve  and  solve  the  resulting  equation 
for  X.  The  roots  of  this  equation  in  x  represent  the  distances  from 
the  origin  to  the  points  of  intersection ;  and  these  distances  are  called 
the  x-intercepts  of  the  given  curve. 

Similarly,  to  find  the  y-intercepts,  put  a;  =  0  in  the  given  equation 
and  solve  the  resulting  equation  for  y. 

Ex.  1.     How  many  x-intercepts  may  a  curve  of  the  nth  degree  have  ?  <^Vv^  ^  >  i^-w^ 
Ex.  2.     What  does  it  mean  when  in  an  equation  in  polar  coordinates  we  put 
^  =  0?    p  =  0? 

26.  A  line  may  be  defined  as  the  path  of  the  moving  point.  Then, 
if  (Xf  y)  be  the  moving  point,  both  x  and  y  are  variable  quantities, 
and  are  called  the  variable  or  current  coordinates  of  the  moving 
point.  The  path  of  the  moving  point  is  then  determined  by  the 
condition  that  its  coordinates  must  vary  only  in  such  a  manner  as 
always  to  satisfy  a  given  equation ;  i.e.  although  both  coordinates  vary 
the  relation  between  them  remains  fixed. 

EXAMPLES 
Find  the  intercepts  and  the  points  of  intersection  of  the  following  loci  : 

1.  2  a;  +  3  y  =  12,        4  a;  -  y  =  5. 

2.  3x  +  5y  =  l,         x-3y+7  =  0. 
8.  5x-22/-f-4  =  0,        x-2y  =  A. 
4.  X  +  3  y  =  15,        x2  +  y2  =  25. 
6.  3  X  -  4  y  =  20,        x^  +  y2  _  10  x  -  10  y  +  25  =  0. 
6.  5x+4y  =  20,        x2  +  ?/2  =  4. 

11.  Find  the  points  of  intersection  of  the  loci  of  Nos.  1,  2,  3,  9,  15,  17,  18,  19, 
20,  21,  26  in  the  last  preceding  set  of  examples. 

12.  Find  the  intercepts  of  the  loci  of  Nos.  7,  9, 10, 11, 12,  13,  14,  18,  19,  20  of 
the  same  set  and  check  tlie  results  by  the  plots  already  made. 

13.  Find  the  area  of  the  triangle  whose  sides  arex  —  3y  +  5  =  0,  3x-|-4y=ll, 
*?  X  +  7  y  =  3. 


7.  x-3y  =  0, 

x2  +  y2  +  20  y  =  0. 

8.  y''  =  4.ax, 

2xy  =  a2. 

9.  y2  =  4ax, 

y2  _  x2  =  a2. 

10.  y^  =  iax, 

x2  =  4  ay. 

34 


v> 


.^ 


LOCI  AND   THEIR  EQUATIONS 


Symmetry  of  Loci 


[27 


27.  The  process  of  constructing  a  locus  explained  in  §  21  is 
long  and  tedious.  It  may  often  be  shortened  by  an  examination 
of  the  peculiarities  of  the  given  equation,  such  as  the  limiting 
values  of  the  variables  for  which  both  are  real  (see  Ex.  3,  §  22), 
symmetry,  etc.  Such  considerations  will  often  reveal  the  general 
form  and  limits  of  the  curve  and  give  all  the  information  desired 
with  little  labor.  The  intercepts  (§  25)  are  almost  always  useful  for 
this  purpose. 

Definitions.  Two  points  A  and  B  are  said  to  be  symmetrical 
with  respect  to  a  ceiitre  0  when  the  line  AB  is  bisected  by  0. 

Two  points  C  and  D  are  said  to  be  syminetrical  with  respect  to  an 
axis  when  the  line  CD  is  bisected  at  right  angles  by  the  axis. 

The  two  points  (x,  y)  and  (—  x,  —  y)  are  symmetrical  with  respect 
to  the  origin ;  (x,  y)  and  (x,  —  y)  with  respect  to  the  avaxis. 

A  curve  is  said  to  be  sym- 
metrical with  respect  to  a  centre 
O  when  all  lines  passing 
through  0  meet  the  curve  in  a 
pair,  or  pairs,  of  points  sym- 
metrical with  respect  to  0. 

A  curve  is  said  to  be  sym- 
metrical with  respect  to  an  axis 
when  all  lines  perpendicular 
to  the  axis  meet  the  curve  in 
a  pair,  or  pairs,  of  points  sym- 
metrical with  respect  to  the 
axis. 

Or,  in  other  words,  a  curve  is  symmetrical  with  respect  to  an 
axis,  if  the  figure  appears  the  same  when  a  plane  mirror  is  placed 
on  the  axis  perpendicular  to  the  plane  of  the  curve. 

The  curve  PQ  is  symmetrical  with  respect  to  the  origin,  and  RS 
is  symmetrical  with  respect  to  the  y-axis. 

The  principal  kinds  of  symmetry  arising  from  the  form  of  the 
equation  are  as  follows : 


28]  LOCI  AND   THEIR  EQUATIONS  35 

28.   Equations  in  Cartesian  Coordinates. 

(1)  If  f{Xy  y)  =/(«,  —  2/)>*  i^^  locus  of  the  equation  fix,  y)  =0  is 
symmUrical  with  respect  to  the  x-axis;  i.e. 

If  an  equation  is  not  altered  when  the  sign  of  y  is  changed,  its  locus 
is  symmetrical  with  respect  to  the  x-axis. 

Let  (x',  y')  be  any  point  on  the  locus  f(x,  y)  =  0. 

Then,  since  f(x,  y)  =f{x,  -  y),  by  hypothesis, 
f(x',y',)=f(x',-y')  =  0. 

That  is,  the  point  (x',  —  y')  is  also  on  the  locus.  Therefore,  since 
the  line  x  =  x'  meets  the  locus  in  any  point  (x',  y'),  it  will  also  meet 
the  locus  in  the  symmetrical  point  (x',  —  y'),  and  the  curve  is 
symmetrical  with  respect  to  the  a;-axis, 

Ex.     Let/(aj,  y)=y^-4x,  thenf(x,  -  y)  =  (-  yy  -  4 x  =  y^  -  i x. 
Therefore /(x,  y,)  =/(x,  —  y)  and  the  curve  ?/2  —  4  x  =  0  is  symmetrical  with 
respect  to  the  x-axis.     (See  Ex.  2,  §  22. ) 

(2)  Similarly,  if  fix,  y)=f(^  —  x,  y),  the  locus  of  f{x,  y)  =  0  is 
symmetrical  with  respect  to  the  y-axis. 

Ex.  y  —  cos  x  =  y  —  cos  (  —  x) . 

Therefore  the  locus  ot  y  =  cos  x  is  symmetrical  with  respect  to  the  y-ajoa. 

(3)  If  f(x,  y)  =  ±f(-  x,  -  y),  the  locus  of  fix,  y)  =  0  is  sym- 
metrical ivith  respect  to  the  origin. 

Let  (x',  ?/')  be  any  point  on  the  locus  f(x,  y)  =  0. 
Then,  since /(»,  y)  ~  ±f(  —  x,  —  y)  by  hypothesis, 

/(a^',2/')=/(-a^',  -.^/)=0. 

Hence  the  straight  line  through  the  origin  and  the  point  (x',  y') 
meets  the  locus  again  in  the  symmetrical  point  (  —  x',  —  y'). 
Therefore  the  curve  is  symmetrical  with  respect  to  the  origin. 

a^     b-^        ~a^         62  a^         6'"* 

=  i-^)\{-yy_i, 

~      a^  b^ 

*  The  sign  "  =  "  means  "  identical  with,"  i.e.  the  same  for  all  values  of  x  and  y,  and 
therefore  that  the  two  expressions  vanish  for  the  same  values  of  x  and  y. 
E.g.     (x  +  y)2=x2+2x2/-f-y2,  eosx  =  cos  (-x). 


36  LOCI   AND   THEIR   EQUATIONS  [28 

Therefore  the  curve  —  +  ^  =  1   is  symmetrical  with  respect  to  both  axes 
or      Jr- 
and  the  origin.  (See  figure,  §  34. ) 

(4)  If  f(x,  y)  f=  {y,  x)  the  locus  off(x,  y)  =  0  is  symmetrical  with 
respect  to  the  line  y  =  x.     E.g.  a^  -f-  ?/^  =  1. 

(5)  If  f{x,y)=f{  —  y,  —  x)  the  locus  of  f{x,  y)  =  0  is  sym- 
metrical with  respect  to  the  line  y  =  —  x.     E.g.  xy  —  ±1. 

Let  the  student  prove  propositions  (4)  and  (5). 

The  foregoing  conditions  of  symmetry  are  both  7iecessary  and 
sufficient;  i.e.  if  either  one  of  the  conditions  (3),  for  example,  is 
satisfied,  the  locus  is  symmetrical  with  respect  to  the  origin,  other- 
wise not.  The  student,  however,  should  examine  the  opposite 
propositions  independently. 

The  following  conditions,  (6),  (7),  (8),  are  sufficient,  but  not 
necessary  ;  i.e.  the  opposite  propositions  are  not  necessarily  true. 

(6)  If  an  equation  contains  only  even  powers  of  y,  its  locus  is  sym- 
metrical with  respect  to  the  x-axis.     [From  (1).] 

(7)  If  an  equation  contains  only  even  powers  of  x,  its  locus  is  sym- 
metrical with  respect  to  the  y-axis.     [From  (2).] 

(8)  If  an  equation  contains  only  even  powers  of  both  x  and  y,  its 
locus  is  symmetrical  with  respect  to  both  axes  and  also  with  respect  to  the 
origin.     [From  (3).] 

In  an  algebraic  *  equation  either  one  of  the  following  conditions  is 
sufficient,  and  one  or  the  other  is  necessary. 

(9)  If  all  the  terms  of  an  algebraic  equation  are  of  even  degree,  or 
if  all  the  terms  are  of  odd  degree,  its  locus  is  symmetrical  with  respect  to 
the  origin.     [From  (3).] 

Show  that  (6),  (7),  (8),  and  (9)  follow  from  (1),  (2),  and  (3). 

Show  that  (6),  (7),  (8)  are  necessary  conditions  of  symmetry  if  the  equation 


*  A  function  in  which  the  variables  are  involved  in  no  other  way  than  by  addition, 
subtraction,  multiplication,  division,  and  root  extraction  is  called  an  Algebraic  Func- 
tion.   All  others  are  called  Transcendental  Functions. 

E.g.    3a;2_2a;  +  4,  x2— aa;?/  +  6r/2,        ^,J  +n  Vxy. 

are  algebraic  functions;   while  a*,  sin  x,  sec-i  y,  log  (x^-j-y)   are  transcendental 
functions. 


(jHj^/l7 


29]  LOCI  AND  THEIR  EQUATIONS  37 

29.   Equations  in  Polar  Coordinates. 

The  best  way  to  determine  the  symmetric  properties  of  loci  in 
polar  coordinates  is  to  transform  their  equations  to  rectangular  co- 
ordinates, and  then  apply  the  tests  given  in  §  28. 

The  following  conditions,  however,  are  useful  in  simple  cases. 
They  are  sufficient  but  not  necessary,  conditions  of  symmetry. 

(1)  If  fid)  =f{-  0),  or,  iffiO)  =  -/(tt  -  6),  the  locus  of  p  =  f{e) 
is  symmetrical  with  respect  to  OX. 

(2)  Similarly,  if  f(e)=f(-7r- 6),  or,  if  f(e)  =  -f(-0),  the  locns 
of  p  =f{0)  is  symmetrical  with  respect  to  O  Y. 

(3)  If  f{6)  =/(7r  +  6),  the  locus  of  p  =f{6)  is  symmetrical  with 
respect  to  O. 

EXAMPLES 

In  what  respects  are  the  loci  of  the  following  equations  symmetrical  ?  Af* 

4  1.   y  =  x2.V>^            2.   y^x^.           y     3.  t  =  ^-         "^4.  y^  =  x.Y 

5.    y  =  7^.                4    6.    x^r=:yx^\         7.  y2^x3.        ^  8.  y^^x^. 

9.   2/2  =  a;2.              -f  10.  y  =  x\  «^        ^       11.  y'^  =  x*.          12.  ^  =  y^- 

13.   y'^  =  xfi.              -^  14.   f  =  x^.    ^'             15.  y^  =  x^.          16.  y"'  =  x*. 

18.    y=x^-x'^.          Ad.  y  =  x*-x^.20.  y  =  x*-3fi. 


y  —  x^  —  X. 

21.  xy  =  a.                 22.   x'^y  =  a.  ^^S.  ax^  +  hy'^  =  \.\^  Vr>^«w^ 

24.  ax2  ^2bxy  +  cy^  =  1.  25.  ax^  +  2bxy-^  ay^  =  1.    ^ 

26.  xy  -  2  (X  +  y)  =  1.  /^7.  x8  +  y3  =  1.  \>  «tMWv 

^^28.  X*  4-  y*  =  l.-l(]^  \sr^H^  29.  x<  =  y2 (4  a^  -  x2).    M 

30.  x(y  +  x)2  +  a^y  =  0S>  31.  xV  =  rt2(x2  +  2/2). 

yZ2.  x^  +  y-'  =  a^.  \j^  ^m^A^  33.  x^  -\- y^  =  a^. 

34.  (a  -  X)  2/2  =  (a  +  x)x2.  ^  35.  (a  -  x)  y2  +  x^  =  0. 

36.  ?/  =  K2'  +  2-*).  37.    y=i(2*-2-^).   38.  p2_cos2^.     39.    p2_sin2^. 

40.  Point  out  the  symmetric  properties  of  the  loci  in  the  last  two  preceding 
sets  of  examples,  especially  those  given  in  polar  coordinates. 

41.  Show  that  if  an  equation  is  not  altered  when  —  x  is  written  in  the  place 
of  y.  and  y  in  the  place  of  x,  its  locus  will  show  no  change  in  position  when  the 
curve  is  turned  about  the  origin  through  a  right  angle  in  its  plane.  For  an 
example  see  No.  7,  p.  27  ;  also  2  x2  —  3  xy  —  2  ?/2  =  ?. 

The  locus  of  x*  +  a'^xy  -  y*  =  0  is  also  such  a  curve. 


38 


LOCI  AND  THEIR  EQUATIONS 


[30 


To  Find  the  Equation  of  a  Locus,  having  given  its  Geo- 
metric Definition 

30.  It  should  be  borne  in  mind  that  to  find  the  equation  of  a  locus 
we  have  merely  to  find  an  equation  that  is  satisfied  by  the  coordi- 
nates of  every  point  on  the  locus,  and  not  satisfied  by  the  coordinates 
of  any  other  point.  It  is  not  easy  to  give  specific  directions  which 
can  be  applied  in  all  cases,  but  the  following  plan  will  be  useful  to 
the  beginner,  at  least  in  the  simpler  cases : 

(1)  Choose  the  system  of  coordinates  best  adapted  to  the  locus 
under  consideration,  and  select  a  convenient  set  of  axes. 

(2)  Write  down  the  geometric  equation  which  expresses  the  given 
geometric  definition,  or  any  known  geometric  property  of  the  locus. 

(3)  Express  this  geometric  equation  in  terms  of  the  chosen  system 
of  coordinates,  and  simplify  the  result. 

The  following  examples  will  give  a  better  idea  of  the  method  of 
procedure  than  any  formal  rules ;  they  should  be  carefully  studied : 


31.    To  find  the  equation  of  any  straight  line. 

Y 


Let  ABC  be  any  straight  line  meeting  the  axes  in  A  and  B. 

Let  OB  =  &,  let  tan  XAO=  m. 

Let  P(xj  y)  be  any  point  on  the  line. 

Draw  PQ  parallel  to  OF,  and  BE  parallel  to  OX 


31]  LOCI   AND   THEIR  EQUATIONS  39 

Then  for  the  geometric  equation  we  have 

QP=QR  +  RP=OB+BR  tan  PBR. 
But         QP  =  yy     OB  =  h,     BR  =  x,     tan  PBR  =  m. 

.'.     y  =  mx  +  b,  (1) 

which  is  the  required  equation. 

For  any  particular  straight  line  the  quantities  m  and  b  remain  the 
same,  and  are  therefore  called  constants.  Of  these,  m,  tlie  tangent 
of  the  angle  between  the  line  and  the  »-axis,  is  called  the  Slope  of 
the  line,  while  b  is  the  ^/-intercept. 

By  giving  suitable  values  to  the  constants  7ii  and  6,  (1)  may  be 
made  to  represent  any  straight  line  whatever,  e.g. 

If  6  =  0,  we  have  ^^^^^^  ^2) 

for  the  equation  of  any  line  through  the  origin. 

Quantities  entering  into  an  equation,  such  as  m  and  //,  which 
remain  constant  so  long  as  we  consider  any  particular  curve,  but 
whose  variation  causes  a  change  in  the  position,  size,  or  shai)e  of  the 
curve,  are  called  Parameters  of  the  curve.* 

Moreover,  any  equation  that  can  be  put  in  the  form  (1),  i.e.  y  equals 
some  multiple  ofx  plus  a  constant,  represents  a  straight  line. 

The  general  equation  of  the  first  degree 

Ax-\-By  +  C=0  (3) 

may  be  written  y=—  —  x—  —  , 

and  therefore  (3)  represents  a  straight   line  whose   slope  is  — — 

C  ^ 

and  whose  y-intercept  is .    (See  §  43.) 

Ex.  1.  If  b  varies  in  (1)  while  m  remains  constant,  how  will  the  line 
change  position  ?  If  m  varies  while  b  remains  constant  ?  If  m  varies 
in  (2)? 

Ex.  2.  What  will  be  true  of  the  signs  of  m  and  b  when  the  line  crosses  the 
various  quadrants  ? 

*  The  difference  between  parameters  and  coordinates  shonld  be  carefully  noted  ; 
also  tb(!  diffi^roiice  in  the  effect  of  a  vsiriation  of  the  parameters  of  an  eqnation  and 
the  variation  of  the  current  coordinates.     (See  §  26.) 


40 


LOCI  AND   THEIR  EQUATIONS 


[32 


32.    To  find  the  equation  of  a  circle  referred  to  any  rectangular  axes. 


Let  r  —  radius,  and  let  C(a,  h)  be  the  centre. 

Let  P(xj  y)  be  any  point  on  the  circle. 

Then  CP  =  r.         [Geometric  equation.] 

But  CP'=  {x  -  af  +  (2/  -  hy.  [(2),  §  7.] 

...     (pc-a)^^{y-l>)^  =  r^  (1) 

is  the  required  equation. 

If  a  =  r  and  6  =  0,  (1)  reduces  to 

aj2  +  2/2-2/'ic=0.  (2) 

If  a  =  —  r  and  6  =  0,  (1)  becomes 

x^^y^  +  2rx  =  Q.  (3) 

y  The  circle  at  the  right  in 

the  figure  is  the  locus  of 
equation  (2) ;  the  circle  at 
the  left  is  the  locus  of  equa- 
tion (3). 

When  the  centre  is  at 
the  origin,  a  =  6  =  0,  and 
we  have  for  the  simplest 
equation    of    the    circle    in 


Cartesian  coordinates  the  standard  form  (§  16), 

aj2  4. 2,2  =  rK 


W 


53]  LOCI  AND   THEIR  EQUATIONS  41 

Moreover,  any  equation  of  the  second  degree  in  which  the  term  in 
xy  is  wanting  and  the  coefficients  of  a^  and  tf  are  equal,  can  be  written 
in  the  form  of  equation  (1),  and  therefore  will  represent  a  circle,  real 
or  imaginary.     For  example,  the  equation 

a;2  +  /-4a;  +  62/-3  =  0 
may  be  written  in  the  form 

(x-2f  +  {y  +  ^f=U, 

which  shows  that  its  locus  is  a  circle  whose  centre  is  at  the  point 
(2,  —  3),  and  whose  radius  is  4. 

EXAMPLES 

1.  What  is  the  form  of  the  equation  and  the  position  of  the  circle,  if  6  =  i  r 
and  a  =  0  ? 

2.  What  are  the  parameters  in  these  equations  ?    Discuss  the  effect  produced 
by  their  variation. 

Find  the  centres  and  radii  of  the  following  circles  : 

5.  a;2^  i/2^2x-4i/  =  0.  6.   a:2  +  2/2-3x  +  5i/  =  0. 

7.  a;2  +  y2_,.6^_4y  ^9^0.*^    -    8.    4(a;2 -f  2/2)_12  x  +  8y  -  23  =  0. 

9.  x2  +  ?/2-|.  (3a;  +  8«/- 11  =0.  10.   4(x2  +  ^z^)- 20x  -  32  ?/ +  25  -  0. 

11.  Find  tue  general  equation  of  a  circle  which  touches  both  axes. 

33.   Polar  equations  of  the  circle. 

It  follows  from  (1),  §  8,  that  the  polar  equation  of  the  circle  whose 
centre  is  at  the  point  (a,  a)  and  whose  radius  is  r,  is 

p2  -2  ap  cos  ((9  -  a)  +  a'  -  /-^  =  0.  (1) 

If  the  pole  is  on  the  circle,  the  equation  is 

p  =  2rcos(^-«);  (2) 

if  the  centre  is  also  on  the  initial  line,  the  equation  is 

p  =  2rcos^;  (3) 

if  the  circle  is  above  the  initial  and  tangent  to  it  at  the  pole,  its 
equation  is  p  =  2  r  sin  0.  (4) 

Ex.  1.  Why  is  (1)  of  the  second  degree  in  p  while  (2),  (3),  and  (4)  are  of  the 
first  degree  ?  When  is  the  pole  outside,  and  when  inside  the  circle  ?  Discuss 
the  effect  of  the  variation  of  the  parameters  in  these  polar  equations. 

Ex.  2.    Transform  equations  (1),  (2),  (3),  (4)  to  rectangular  coordinates. 


42 


LOCI  AND  THEIR  EQUATIONS 


[34 


34.  The  Ellipse.  The  ellipse  is  the  locus  of  a  point  which  moves 
so  that  the  sum  of  its  distarices  from  two  fixed  points,  called  foci,  is  con- 
stant. 


Take  the  line  through  the  foci  as  the  a>axis,  and  the  point  midway 
between  the  foci  as  origin. 

Let  2  a  =  the  sum  of  the  distances  from  any  point  on  the  ellipse 
to  the  foci.     Let  F{cj  0)  and  F\—c,  0)  be  the  two  foci. 

Let  P{x,  y)  be  any  point  on  the  locus. 

Then  FP  +  F^P=  2  a.  [Geometric  equation.] 

But  FP  =  ^  (x-cy  +  y\ 

and 


FP  =  -\/  {x-^cf  +  y\ 
Transposing  the  first  radical  and  squaring 


[(2),  §  7.] 
2  a.  (1) 


(oj  +  cf +  2/'  =  4a2+(a;-c)24.2/2-4a  V(a;-c)2  +  /, 


or  a  V  (a;  —  c)^  +  y-  ==  a^  —  ex. 

Squaring  and  transposing  again 

{a^  _  c2)  a^  +  ay  =  a^a"  -  c"). 
If  we  put  a^  —  c^—b^,  we  get  the  equation  of  the  ellipse  in  the 


standard  form, 


a* 


62 


(2) 


35]  LOCI  AND   THEIR  EQUATIONS  43 

35.  An  examination  of  this  equation  (2)  as  to  symmetry,  limiting 
values  of  the  variables  and  intercepts,  will  give  the  general  form  and 
limits  of  the  curve. 

(1)  Only  the  square  of  the  variables  x  and  y  appear  in  this 
equation. 

Therefore  the  ellipse  is  symmetrical  with  respect  to  both  axes, 
and  also  with  respect  to  the  origin.    [(8),  §  28.] 

Hence  every  chord  passing  through  O  is  bisected  by  0.  For  this 
reason,  the  point  O  is  called  the  Centre  of  the  ellipse.  Likewise 
the  lines  AA^  and  BB^  are  called  the  Major  Axis  and  Minor  Axis, 
respectively. 

(2)  When  yz=0,x=±a,  a>-intercepts. 
When  a;  =  0, 2/  =  ±  6,  ^/-intercepts. 

Therefore  the  curve  cuts  the  «-axis  a  units  to  the  right  and  a  units 
to  the  left,  the  ?/-axis  b  units  above  and  h  units  below  the  origin. 

(3)  Solving  the  equation  (2)  for  y  and  x  respectively  we  find 
h    , .„  a 


r^^ 


Hence  y  is  imaginary  when  a;  >  a,  or  a;  <  —  a ;  and  x  is  imaginary 
when  2/  >  6,  or  2/  <  —  &. 

Therefore  the  curve  lies  wholly  within  the  rectangle  formed  by 
the  lines  x=  ±  a. and  y=  ±b. 

Also,  as  either  variable  increases,  the  other  diminishes.  The  form 
of  the  curve  is  shown  in  the  figure. 

Such  an  examination  of  an  equation  is  called  A  Discussion  of  the 
Equation. 

Ex.  1.   Transform  equation  (2),  §  34,  to  polar  coordinates  and  show  that  p  is 

finite  for  all  values  of  6. 

Ex.  2.    Where  is  the  point  (h,  k)  if -  +  ^-l>0?     < 0 ? 

Ex.  3.  Show  the  relation  of  the  ellipse  ^^  +  ^  =  1  to  the  circles  x^  +  y^  =  a* 
and  x'^  +  y^  =  62.  «       ^ 


/ 


X.  4.  Find  the  axes,  coordinates  of  the  foci,  and  plot  the  ellipses 


44  LOCI  AND   THEIR  EQUATIONS  [37 

36.  The  Hyperbola.  The  Jiyperbola  is  the  locus  of  a  point  which 
moves  so  that  the  difference  of  its  distances  from  two  fixed  points  {foci) 
is  constant. 

Choose  axes  as  in  the  case  of  the  ellipse,  let  2  a  be  the  constant 
difference,  and  show  that  when  b^=  c^—  a^  the  equation  of  the  hyper- 
bola reduces  to  the  standard  form.     [See  Fig.  §  90.] 

Ex.  1.    Discuss  equation  (1). 

Ex.  2.  Show  that  the  hyperbola  (1)  lies  wholly  between  the  two  straight 
lines  ay  =  ±i  bx,  and  that  as  x  becomes  infinite  the  ordinates  of  the  lines  become 
equal  to  the  ordinates  of  the  hyperbola.  These  lines  are  called  the  Asymptotes 
of  the  hyperbola.     [See  Fig.  §  110.]  C--     r    4.  ':  v 

Ex.  3.   Transform  equation  (1)  to  polar  coordinates,  and  find  the  value  of  p 

b 
when  ^  =  ±  tan-i -• 

Ex.  4.    Find  the  foci,  equations  of  the  asymptotes,  and  trace  the  curves 

^  ^  16      9  ^  ^   16     25  ^  ^    4       16 

^    (4)  a;2  -  if  =  a2.  (5)  tf  -  x'^  =  h\  (6)  4  x2  -  ^/2  =  4. 

37.  The  Parabola.  Tlie  parabola  is  the  locus  of  a  point  whose 
distance  from  a  fixed  straight  line  is  equal  to  its  distance  from  a  fixed 
point. 

The  fixed  point  is  called  the  Focus;  the  fixed  line  is  called  the 
Directrix. 

Take  the  line  through  the  focus  perpendicular  to  the  directrix  as 
the  a^axis,  and  the  origin  midway  between  the  focus  and  the  direc- 
trix ;  let  2  a  denote  the  distance  from  the  focus  to  the  directrix. 
[See  Fig.  §  88.] 

Then  show  that  the  equation  of  the  parabola  is 

2/2  =  4  ax.  (1) 

Ex.  1.   Discuss  this  equation  (1),  also  2/2  =  —  4  ax  and  x!^  =  ±4  ay. 
Find  the  foci,  equations  of'the  directrices,  and  draw  the  parabolas 

(2)  2/2  ^  4  a;. .  (3)  ^2  ^  _  8  X.  (4)  y^  =  Qx. 

(5)  a;2  =  8  2/.  (6)  x2  =  -  10  j/.  (7)  x2  =  -  12  2/. 


37]  LOCI  AND   THEIR  EQUATIONS  45 

EXAMPLES 

I       1.    A  moving  point  is  always  four  times  as  far  from  the  x-axis  as  from  the  i/  -s   ^ 
y-axis.     What  is  the  equation  of  its  locus  ?  A 

2.   Find  the  locus  of  a  point  which  is  equidistant  from  the  two  points  (3,  2)    \y^ 
and  ( -  2,  1).  Ans.     6  x  +  2/  =  4. 

Y-      3.   Find  the  locus  of  a  point  which  is  equidistant  from  the  points  (a,  6)    ^ 
and  (c,  d). 

4.   A  point  moves  so  that  its  distance  from  the  point  (3,  —  4)  is  always  5.  ^^"^ 
Find  the  equation  of  its  locus.     Does  the  locus   pass  through  the  origin  ? 
Why?  Ans.     a;2  ^  y2  -  6x  -f-  8 y  =  0. 

f-        6.   Find  the  equation  of  a  circle  touching  both  axes  and  having  its  centre 
at  that  point  (  — 3,  3). 

6.    Find  the  equation  of  a  circle  touching  both  axes  and  having  a  radius 
equal  to  4. 

"f  7.  A  point  P  is  two  units  from  a  circle  with  radius  4  and  centre  at  (2,  —  6). 
What  is  the  locus  of  P? 

8.   A  point  moves  so  that  its  distance  from  the  origin  is  twice  its  distance 
from  the  x-axis.     What  is  the  equation  of  its  locus  ?  Ans.     x^  —  3  y2  _  q. 

-r  9.  A  point  moves  so  that  its  distance  from  the  x-axis  is  equal  to  its  dis- 
tance from  the  point  (2,  —  3).  Show  that  the  equation  of  its  locus  is 
x2-4x  +  6y  +  13  =  0. 

10.  A  point  P  moves  so  that  its  distances  from  the  points  A  (2,  2)  and 
5  ( —  2,  —  2)  satisfy  the  condition  AF  +  PP  =  8.  Show  that  the  equation  of 
its  locus  is  3  x2  -  2  x?/  +  3  ?/2  =  32. 

11.  What  is  the  locus  of  a  point  which  moves  so  that  (1)  the  sum,  (2)  the 
difference,  (3)  the  product,  (4)  the  quotient  of  its  distances  from  the  axes  is 
constant  (a)  ? 

12.  What  is  the  locus  of  a  point  which  moves  so  that  (1)  the  sum,  (2)  the 
difference,  (3)  the  product,  (4)  the  quotient  of  the  squares  of  its  distances  from 
the  axes  is  constant  (a^)  ? 

13.  Find  the  locus  of  a  point  which  moves  so  that  the  sum  of  the  squares     \^ 
of  its  distances  from  the  points  (a,  0)  and  (—  a,  0)  is  constant  (2  c^). 

14.  Find  the  locus  of  a  point  which  moves  so  that  the  sum  of  the  squares 
of  its  distances  from  the  three  points  (5,  -  1),  (3,  4),  (-2,  -  3)  is  always  64. 

16.  Find  the  locus  of  a  point  which  moves  so  that  the  difference  of  the 
squares  of  its  distances  from  (a,  0)  and  (—  a,  0)  is  the  constant  2  c^. 

"^      16.   Find  the  locus  of  a  point  such  that  the  sum  of  the  squares  of  its  distances   ^ 
from  the  sides  of  a  square  is  constant. 


CHAPTER   III 


THE  STRAIGHT  LINE 


38.  It  was  shown  in  §  31  that  the  equation  of  any  straight  line 
when  expressed  in  terms  of  its  slope  m  and  ^/-intercept  h  is  an 
equatiion  of  the  first  degree, 

y  =  mx  +  6 ; 
and  also  that  the  general  equation  of  the  first  degree, 

Ax-^By-\-C  =  % 
represents  a  straight  line.     It  is  sometimes  more  convenient,  how- 
ever, to  write  the  equation  of  the  straight  line  in  other  forms ;  i.e. 
to  express  it  in  terms  of  some  other  pair  of  parameters. 

39.  To  find  the  equation  of  the  straight  line  in  terms  of  its  inter- 
cepts on  the  axes. 

Y 


Let  A  and  B  be  the  points  in  which  the  straight  line  meets  the 
axes  ;  let  OA  =  a,  and  OB  =  h.    Let  P  (x,  y)  be  any  point  on  the  line. 
Draw  PQ  parallel  to  the  ?/-axis,  and  join  0  and  P. 
Then  A  OAP  +  A  OBP  =  A  OAB. 

Hence "  bx-\-ay  =  ab. 


or 


a     o 


(1) 


40] 


THE   STRAIGHT  LINE 


47 


If  Z  =  -  and  m  =  j,  the  equation  may  be  written 

lx-{-my  =  1.  (2) 

40.  To  find  the  equation  of  a  straight  line  in  terms  of  the  length  of 
the  perpendicular  from  the  origin  iqyon  the  line  and  the  angle  which  that 
perpendicular  makes  with  the  x-axis. 


Let  ON  be  perpendicular  to  the  straight  line  AB,  and  intersect  it 
in  R.     Let  OR  =p,  and  angle  XON=  a. 

Let  P(x,  y)  be  any  point  on  the  line. 

Then  since  OQPR  is  a  closed  polygon,  OR  is  equal  to  the  sum  of 
the  projections  of  OQ,  QP,  and  PR  upon  OR.     That  is, 

022  =  proj.  of  OQ  +  proj.  of  QP+proj.  of  PR 

=  OQ  cos  a  +  QP  sin  a  +  0. 

.*.  occo^a +  ymna=p^  (1) 

which  is  the  required  equation. 

Let  Z  X AP  =  y  =  90°  -^  a.  Then  cos  «  =  sin  y,  sin  a  =  — cos  y, 
and,  by  substituting  in  (1),  the  equation  of  the  line  becomes 

a5  sin  -y  -  2/  cos  -y  =  p,  (2) 

Since  equations  (1)  and  (2)  involve  the  trigonometric  functions,  sin 
and  cos,  ON  and  AB  must  be  regarded  as  directed  Hues.  As  in 
Trigonometry,  we  will  consider  the  directions  of  the  terminal  lines 
of  a  and  y  as  the  positive  directions  of  these  lines. 

If  y  =  90°  4- «,  as  assumed  above,  then  standing  at  R  facing  the 
positive  direction  of  ON,  the  positive  direction  of  AB  is  to  the  lefl; 


48 


THE   STRAIGHT   LINE 


[40 


and  standing  at  R  facing-  the  positive  direction  of  ABj  the  positive 
direction  of  ON  is  /rom  AB  toivard  the  right. 

This  will  be  called  the  positive  side  of  the  line  AB. 

Then  in  equations  (1)  and  (2)  p  is  positive  when  taken  in  the 
positive  direction  of  ON.  Hence  when  p  is  positive  the  origin  is  on 
the  negative  side  of  the  line. 


U.g.     In  the  equations 

>y^'^ 

cos  a  =  sina  =  — . 

V2 

/.  a  =  45°  and  y  =  135° 

for  both  lines  ;  but 

for  AB           p  =  3, 

for  CD           p=-S. 

Hence  the  two  lines  are  parallel  but 
on  opposite  sides  of  0.    Also  0  is  on 
the  positive  side  of  CD  and  on  the 
negative  side  of  AB. 

Since  sin  (^  ±  7r)  =  —  sin  ^  and  cos  (^  ±  7r)  =  —  cos  0, 

if  the  signs  of  all  the  terms  in  (1),  or  (2),  be  changed,  the  direction 
of  AB,  and  also  of  ON,  will  be  changed  by  ±  tt  ;  and  therefore  the 
positive  and  negative  sides  of  the  line  will  be  reversed.  That  is, 
the  equation  of  a  line  may  be  written  so  as  to  make  either  side  of 
the  line  positive  or  negative,  just  as  we  choose. 

E.g.     The  equation  of  the  line  AB, 

2         2 
may  also  be  written 

X      VSy 
2         2 
In  (1)  p  : 


-2, 


2. 

-2, 


cos  ct  =  sin  7  =  -, 

2' 


sni  a 


Vs 

—  cos  7  = . 


\a  =  -  60°  and  y  =  30°. 


41] 


THE'  STRAIGHT  LINE 

1 


49 


In  (2)         i?  =  2,    cos  a  =  sin  y 

.  •.  a  =  120°  and  7  =  210^ 


^,    sina  =  —  CO87  =A^. 
2  2 


Angles  and  directions  corresponding  to  (1)  are  denoted  by  single  arrow-heads, 
those  corresponding  to  (2)  by  double  arrow-heads. 

The  origin  is  on  the  positive  or  negative  side  of  the  line  according  as  the 
equation  is  written  in  the  form  (1)  or  (2). 

Ex.  Point  out  the  combinations  of  signs  of  cos  «,  sin  a,  and  p  when  the  line 
crosses  the  different  quadrants. 

41.     Transformation  of  the  equations  of  the  straight  line. 

In  §§  31,  39,  and  40  we  have  found,  by  independent  methods,  the 
three  standard  forms  of  the  equation  of  a  straight  line  involving 
different  pairs  of  parameters,  m  and  h,  a  and  &,  a  or  y,  and  p ;  viz. : 

y  =  mx  +  b.    Slope  form,  (1) 


-  +  ?  =  1 ,     Intercept  form, 


(2) 


"i       •  ^.^r.^        Z  c     Distance,  or  normal  form.  (3) 

i X  smy  -  y  cos y  =  p, )  '  *'  ^  ' 

Any  one  of  these  forms  of  the  equation  may,  however,  he  deduced 
from  any  other. 


I.   From  the  figure  we  obtain  di- 
rectly the  relations 

cos  a  _      b 

~      a 


,  sin  y 

m  =  tan  y  =  — — ^  = 


cos  y  sin  a 

and        j9  =  a  cos  a  =  6  sin  a 

=  —  6  cos  y  =  a  sin  y. 

Then  substituting  these  values  of  m  in  (1),  for  example,  gives 


cos« 
sin  a 


x-[-b, 


and 


smy 
^       cos  y      ' 


50  THE   STRAIGHT   LINE  [41 

Whence,  since  6  sin  a  =  —h  cos  y  =  p,  we  get 

a;  COS  ot  H- 2/ sin  ct  =^, 
and  X  sin  y  —  yGOSy=p. 

Moreover,  the  general  equation  of  the  first  degree, 

Ax  +  By+C  =  0,  (4) 

can  be  transformed  into  any  one  of  the  three  standard  forms. 

II.  Solving  (4)  for  y  gives  (see  §  31) 

A        O 
y=  --X--.     Slope  form.  (5) 

a       Jo 

III.  If  we  transpose  and  divide  by  C,  (4)  may  be  written 

— TvH p=l.     Intercept  form.  (6) 

^A     ~B 

IV.  To  reduce  the  general  equation  (4)  to  the  distance  form. 

In  this  case  we  are  to  transform  (4)  so  that  the  sum  of  the  squares 
of  the  resulting  coefficients  of  x  and  y  shall  be  unity.  Hence,  if  we 
assume  the  transformed  equation  to  be 

KAx  +  KBy  +  KG  =  0,  (7) 

then  K^A^  +  KV-  =  cos^  a  +  sin^  a  =  1. 

j^^         1 
Whence  V3M^^ 

.-.  ^       x-^        ^       y= ^ (8) 

V^^  +  52       VA'-^B'  VA'+& 

is  the  required  equation. 

Hence,  to  reduce  the  general  equation  (4)  to  the  distance  form,  trans- 
pose C  and  divide  by  ■\/ A^  +  B^. 

The  general  equation  of  the  first  degree  must  therefore  represent 
a  straight  line,  since,  by  transposing  .and  multiplying  by  a  suitable 
constant,  it  can  be  reduced  to  any  one  of  the  standard  forms  of  the 
equation  of  the  straight  line.     {Cf  §  31.) 


41]  THE   STRAIGHT   LINE  51 

V.    Values  of  parameters  in  terms  of  A,  B,  and  C. 

Comparing  coefficients  in  (1)  and  (5),  (2)  and  (6),  (3)  and  (8),  we 

get  a  =  -^»    ^  =  -^'    m  =  -^,  p=       ~^ 


A'  B'  B'  ^       ^W+B' 

A  B 

cos  a  =  sin  v  =  —  .    sin  a  —  —  cos  v  = 


V^2  +  ^  V^-  +  B' 

Observe  that  the  values  of  a  and  h  thus  obtained  are  the  same 
as   those   found   by  putting   y  =  0,  then   x  —  0  in    (4) ;    also   that 

A          b 

m  =  — —  = ,   as   found   above   directly  from  the  figure.     Then 

B         a 

sin  a,  cos  a,  and  p  can  be  found  by  Trigonometry  and  the  relations 

obtained  from  the  figure. 

EXAMPLES 

1.    When  is  it  impossible  to  write  the  equation  of  a  straight  line  in  the 
intercept  form  ?    in  the  slope  form  ? 

Change  the  following  equations  to  the  standard  forms  and  thus  determine 
their  parameters.     Also  draw  the  lines : 

3.  4?/ =  3  a; +  24. 

6.  5x+4«/  =  20. 

7.  2x-4y  +  9  =  0. 
9.  2x  +  3y  =  0. 

11.    y  =  4. 

Transform  Ax  ■]-  By  ^  C  =  0  so  that  the  sum  of  the  three  coefficients 
shall  be  if;  so  that  the  square  of  the  first  shall  be  three  times  the  second  ;  so 
that  the  product  of  the  three  shall  be  twice  their  sum. 

13.  Transform  5  a;  +  4  y  —  20  =  0  so  that  the  sum  of  the  three  coefficients 
shall  be  22  ;  so  that  the  product  of  the  first  and  third  shall  be  equal  to  the  second. 

14.  Transform  3a;-4?/+12  =  0so  that  the  square  of  the  second  coefficient 
shall  be  equal  to  twice  the  third  minus  four  times  the  first ;  so  that  the  product 
of  the  three  shall  be  minus  three  times  the  last. 

16.  Transform  5  a;  —  2  y  —  3  =  0  so  that  the  product  of  the  first  and  second 
coefficients  minus  ten  times  the  third  shall  be  equal  to  —  40  ;  so  that  the  s(iuare 
of  the  second  plus  twice  the  sum  of  the  first  and  third  shall  be  equal  to  24. 


2. 

x-\-V3y  +  10  =  0. 

4. 

y  =  x-6. 

6. 

5  X  -  12  y  =  13. 

8. 

2x-Sy  =  i. 

10. 

x-a  =  0. 

12. 

Transform  Ax  +  B 

\}^ 


52  THE  STRAIGHT  LINE  [42 

42.    To  jind  the  polar  equation  of  a  straight  line. 


Let  N{p,  a)  be  the  foot  of  the  perpendicular  from  0  upon  the 
given  line  AB. 

Let  P(p,  6)  be  any  other  point  on  AB. 

Then  Z  NOP  =  {6  -  a), 

and  OP  cos  JV'OP  =  ON. 

.•.pcos(e-a)=i>,  (1) 

which  is  the  required  equation. 

EXAMPLES 
Find  the  parameters  and  draw  the  lines  whose  equations  are 
1.   p  cos  {6  -  30°)  =2.  2.    p  cos  (d  -  60°)  =  1. 

3.   p  cos  {d  +  45°)  =3.  4.    p  cos  {6  +  120°)  +4  =  0. 

5.    p  cos  {e  -  120°)  +1=0.  6.    p  cos  {6  +  60°)  +  5  =  0. 

<    7.   Transform  aj  cos  a  +  y  sin  a  =  j?  to  polar  coordinates. 

8.    What  is  the  polar  equation  of  a  line  perpendicular  to  the  initial  line  ? 
parallel  to  the  initial  line  ? 

•^9.    What  is  the  polar  equation  of  any  straight  line  through  the  pole  ?   of  the 
initial  line  ? 

10.  What  locus  is  represented  by  sin  ^  =  0  ?     sin  2  ^  =  0  ?     sin  3  ^  =  0  ? 
•..  sin  w5  =  0  ? 

11.  What  is  the  locus  of  cos  nO  =  0  when  w  =  1,  2,  3,  ...  ? 

12.  Find  the  coordinates  of  the  point  of  intersection  of  p  cos  {6  ±  45°)  =  1. 

J    13.    Find  the  polar  equations  of  the  bisectors  of  the  angles  between  the  lines 
p  cos  (d  -  60°)  =  2,   and  p  cos  (J9  -  30°)  =  2. 


V 


V^:  ■ 


44]  THE   STRAIGHT   LINE  53 

43.  To  find  the  equation  of  a  straight  line  passing  throufjh  a  fixed 
point  {xij  t/i)  in  a  given  direction. 

Let  the  line  make  with  the  a^axis  an  angle  tan~^  m. 
Its  equation  will  then  be  (where  b  is  unknown)  _     _ 

y  =  mx-\-bj  (1) 

and  since  the  line  passes  through  (x^,  ?/i), 

yi  =  mxi-\-b.  (2) 

Whence,  by  subtracting  (2)  from  (1), 

y-yi  =  m{ac-i€i).  (3) 

The  line  given  by  (3)  will  pass  through  the  point  (xy,  y^  for 
all  values  of  m\  and  may  be  made  to  represent  any  line  through 
{xiy  ?/i)  by  giving  to  m  a  suitable  value. 

If  then  we  know  a  line  passes  through  a  certain  point,  we  may 
write  its  equation  in  the  form  (3),  and  determine  the  value  of  m 
from  the  other  condition  the  line  is  made  to  satisfy. 

Since  m  =  tan  y  =  ^(§  40),  equation  (3)  may  be  written   in 

the  form 

^^^im^iLzJll^r,  (4) 

cos-y         sinY  ^  ^ 

where  r  is  the  variable  distance  from  the  fixed  point  (x^,  y^  to  any 
point  (x,  y)  on  the  line.     This  is  a  very  useful  formula. 
Let  the  student  prove  (4)  directly  from  a  figure. 

44.  To  find  the  equation  of  a  straight  line  ivhich  passes  through  two 
given  points  (x^  2/1)  ci'^d  (x2,  y^. 

Since  the  line  passes  through  (ajj,  2/1),  its  equation  will  be  of  the 
form  [(3),  §  43] 

y-y^  =  m(x-Xi)',  (1) 

then,  since  (ajg,  ^2)  is  also  on  the  line,  we  have 

y2-yi='m(x2-Xi).  (2) 

Dividing  (1)  by  (2)  gives  the  required  equation 


2/2  -  2/1      iK2  -  «! 


(3) 


54 


THE   STRAIGHT  LINE 


[45 


Equation  (3)  may  also  be  written 

X,    y,  1 
xu  2/1,  1=0.  (4) 

which  is  obvious,  since  the  area  of  the  triangle  formed  by  (fl?i,  2/1) 
(x2, 2/2)  and  any  other  point  (x,  y)  on  the  line  is  zero. 


EXAMPLES 

Find  the  equation  of  the  straight  line 
1.   if  6  =  f  and  7  =  tan-i  \. 


/a.    if  a  = 


h  and  p 


3.   if  7  =  30°  and  p  =  4.  4.    if  6  =  -  3  and  7  =  150°. 

6.    if  7  =  tan-i  2  and  the  line  passes  through  (3,  —  4). 

6.  if  7  =  tan-i  -  and  the  line  passes  through  (—a,  h). 

7.  passing  through  the  pairs  of  points  (2,  3)  and  (—  6,  1)  ;  (-  1,  3)  and 
(6,  -  7)  ;  (a,  6)  and  (a  +  6,  a-  b). 

8.  Find  the  equations  of  the  sides  of  the  triangle  whose  vertices  are  the 
points  (1,  3),  (3,  -  5),  and  (-  1,  -  3). 

9.  Find  the  equations  of  the  three  medians  of  this  triangle,  and  show  that 
they  meet  in  a  point. 

10.    Find  the  equation  of  a  line  passing  through  (—1,  4)  and  having  inter- 
cepts (1)  equal  in  length,  (2)  equal  in  length  but  opposite  in  sign. 
^  11.   What  is  the  equation  of  the  line  through  (4,  —  5)  parallel  to2x4-32/  =  6? 

45.    To  find  the  angle  between  two  straight  lines  whose  equations  are 
given. 


Let  AB  and  A'B'  be  the  given  lines. 


45]  THE   STRAIGHT   LINE  55 

Let  <f>  be  the  required  angle. 

Then,  using  the  same  notation  and  the  same  convention  as  to 
direction  of  the  lines  as  in  §  40, 

<j>  =  a-a'  =  y-y'.  ~  (i) 

I.  If  the  equations  of  the  given  lines  be 

X  cos  a-\-y  sin  a  =p     and     x  cos  a'  -\-y  sin  a'  =i>', 
cos  </)  can  be  found  by  direct  substitution  in 

cos  <^  =  cos  a  cos  a'  +  sin  a  sin  a'.  (2) 

II.  If  the  equations  of  the  given  lines  be 

y  =  mx-\-b     and    y  =  m'x  +  b', 

we  have  from  (1),  since  tan  y  =  m,  and  tan  y'  =m', 

.       ,      .       /         ,x         tan  y  —  tan  y'         m  —  m'  ,^. 

tan  d)  =  tan  (y  —  7')  = :: -^ '—,  = ,.  (3) 

^  ^'^      ^^      1  +  tanytany'     l+mm'  ^^ 

When  m  =  m\  tan  <^  =  0,  and  the  lines  are  parallel. 
When  1  +  mm)  —  0,  tan  <^  is  infinite. 

Therefore,  when  wi'= ,  the  lines   are   perpendicular    to    one 

another. 

III.  If  the  equations  of  the  lines  be 

^a;  +  %H-C=0     and     ^'x-f  B'l/ +  C'  =  0, 

A  A' 

then     m  =  -  — ,     m'  =  -  — ;     and  therefore,  from  (3), 

If  ^'JB  -  AB'  =  0,  ie.  if  ^,  =  ^,  the  lines  will  be  parallel. 

If  AA' -\- BB' =  Of  the  lines  will  be  at  right  angles  to  one 
another. 

It  should  be  noticed  that  (2)  gives  the  angle  between  two  di- 
rected lines.  For  if  all  the  signs  in  one  of  the  equations  in  I  be 
changed,  the  direction  of  the  line  will  be  changed  by  ±  tt,  (§  40). 


56  THE   STRAIGHT   LINE  [46 

The  sign  of  cos  <^  given  by  (2)  will  also  be  changed  and  <^  becomes 
the  supplement  of  its  former  value.  But  if  all  the  signs  in  both 
equations  be  changed,  <^  is  unaltered. 

If  the  equations  be  so  written  that  the  origin  is  on  the  sa7ne  side 
(either  positive  or  negative)  of  both  lines,  it  will  be  in  the  obtuse 
angle  between  the  lines  when  cos  <^  is  positive,  and  in  the  acute 
angle  when  cos  <^  is  negative. 

If  m  and  m'  be  so  taken  that  m'  >  m,  then  y'  >  y  and  (3)  will 
give  tan  (—  <^)  =  —  tan  <^  =  tan  (tt  —  <^),  instead  of  tan  <^. 

46.  To  find  the  equations  of  two  lines  passing  through  a  given  point 
(xi,  yi)j  the  one  parallel,  the  other  perpendicular,  to  a  given  line. 

Let  the  given  line  be 

Ax  +  By+C  =  0. 

Then  the  parallel  line  is 

Ax-\-By  +  K==0,  [§45,  III.]     (1) 

and  the  perpendicular  line  is 

Bx-Ay  +  K'  =  0,  [§  45,  III.]     (2) 

where  K  and  K'  are  constants  to  be  determined. 

Since  both  (1)  and  (2)  are  to  go  through  (xi,  y^),  these  constants 
are  such  that 

Ax,-^By,-]-K=0   I  ^3^ 

and  Bxi  —  ^?/i  +  ^  =  0,  J 

i.e.  K=-(Ax,  +  By,)  |  ,^. 

and  K'  =  -(Bxi-Ay,).] 

Therefore,  the  required  equations  are,  respectively, 

A(ac-iet)  +  B{y-  y{)  =  0,  (5) 

and  Bix-xt)-A(y-yi)  =  0.  (6) 

If  the  equation  of  the  given  line  is  in  the  form 
y  =  mx  +  b, 
the  required  equations  may  be  written  [(3),  §  43,  and  II,  §  45] 

y-yi  =  m(i€-xi)  (7) 

and  V-yi=^-^(^-oci)»  (8) 


46]  THE  STRAIGHT   LINE  67 

EXAMPLES 

Find  the  angles  between  the  following  pairs  of  lines : 

1.  3  a;  +  4  ?/  =  8  and  7  y  -  X  +  14  =  0. 

2.  2a;  +  3|/  =  6  and  2y  =  3x- 12. 
^     3.   x  +  4  =  2y  and  x  +  Sy  =  9. 

4.   3  2/  +  12  a;  +  16  =  0  and  2  y  =  4  a;  +  5. 

^.      5.    --f  =  1  and  ^--  =  1. 
^  a     b  ah 

-|L      6.   Prove  that  the  points  (1,  3),  (5,  0),  (0,  -  4),  and  (-4,  -  1)  are  the 
vertices  of  a  parallelogram,  and  find  the  angle  between  its  diagonals. 

Find  the  equations  of  the  two  straight  lines 

7.   passing  through  the  point  (2,  3),  the  one  parallel,  the  other  perpendicular, 
to  the  line  4  x  —  3  y  =  6. 

^  8.  passing  through  (4,  —  2),  the  one  parallel,  the  other  perpendicular,  to  the 
line  y  =  2  X  +  4. 

J  9.  passing  through  the  intersection  of4x  +  y  +  6  =  0  and  2x  —  3y  +  13  =  0, 
one  parallel,  the  other  perpendicular,  to  the  line  through  the  two  points  (3,  1), 
and  (-1,  -2). 

.^,  10.  Find  the  equation  of  the  perpendicular  bisector  of  the  line  joining  the 
points  (3,  -1)  and  (-2,  1). 

i  11.  Find  the  equations  of  the  lines  perpendicular  to  the  line  joining  (2,  1) 
and  ( —  3,  —  2)  at  the  points  which  divide  it  internally  and  externally  in  the 
ratio  2 : 3. 

12.  What  is  the  equation  of  a  line  parallel  to  3  x  +  4  y  =  12  and  at  a  distance 
4  from  the  origin '? 

13.  Find  the  point  of  intersection  of  two  parallel  lines.      (eP,  o^] 
The  vertices  of  a  triangle  are  (3,  1),  (-  2,  3),  and  (2,  -  4): 

14.  Find  the  equations  of  its  altitudes  and  show  that  they  meet  in  a  point. 

15.  Find  the  equations  of  the  perpendicular  bisectors  of  its  sides,  and  show 
that  they  meet  in  a  point  which  is  equidistant  from  the  three  vertices. 

16.  Find  its  interior  angles. 

17.  Find  the  equations  of  two  lines  through  the  origin,  each  making  an  angle 
of  30°  with  the  line  4  x  -f  ?/  +  4  =  0. 

/  18.  Show  that  the  equations  of  the  two  straight  lines  through  a  given  point 
(3Ci,  J/i)  making  a  given  angle  0  with  the  line  y  =  mx  +  b  are 

m  i-  tan  0  , 


58 


THE  STRAIGHT  LINE 


[47 


47.    To  find  the  perpendicular  distance  from  a  given  straight  line  to  a 
given  point  Px{x^y  2/i)- 


Let  HK  be  the  given  line,  and  let  H^K^  be  parallel  to  HK  and 
pass  through  Pj.     Let  P^Q  be  the  perpendicular  from  P^  on  HK^ 
and  OR,  OE'  the  perpendiculars  from  0  on  HKsind  H'K'. 
Let  the  equation  of  HK  be 

a;  cos  a-\-y  sin  a=p. 
Then  the  equation  of  WK  is 

a;  cos  a  +  2/  sin  a  =  p  +  RR  =  p  +  QPi ; 
and  since  this  line  (2)  goes  through  Pi(x^,  y^, 

fljj  cos  a  +  2/i  sin  a=p  -\-  QP^. 
.'.  QP\  =  oc\  cos  a  +  2/1  sin  a  -  p, 

which  is  the  distance  from  the  line  a,  p  to  the  point  (x^y  2/1). 
If  the  equation  of  the  given  line  be 

Ax  +  By-\-C  =  0, 
A  .  B  -0 


(1) 

(2) 

(3) 
(4) 


cos  a  = 


V  ^^  +  B' 


sina  = 


■Va'  +  b' 


i>  = 


■Va'  +  b' 


[§  41,  v.] 


and  substituting  these  values  in  (4)  gives 

Aact  +  Byi  +  C 

which  is  the  distance  from  line  A,  B,  C  to  the  point  (x^,  y^. 


(5) 


48]  THE  STRAIGHT  LINE  59 

Hence  the  length  of  the  perpendicular  from  a  given  line  to  a  given 
point  is  found  by  substituting  the  coordinates  of  the  point  in  the  equa- 
tion of  the  line  reduced  to  the  distance  form  with  all  the  terms  trans- 
posed to  the  first  member. 

The  expression  (5)  will  be  positive  or  negative  accordrng-  as 
Ax^  +  Byi  -f  C  is  positive  or  negative  (if  V^^  +  J5^  be  positive).  If 
Ax^  +  By^  -f  C  is  positive,  the  point  {x^,  t/i)  is  said  to  be  on  the  positive 
side  of  the  line  Ax  +  By  +(7=0;  if  Ax^  -f-  By^  +  C  is  negative, 
(^1?  2/i)  is  said  to  be  on  the  negative  side  of  the  line.  If  the  equation 
of  the  line  be  written  so  that  p  is  positive,  the  expression  (5)  will  be 
found  to  be  positive  when  Pj  and  0  are  on  opposite  sides  of  the  line. 
{Cf  §  40.) 

Hence  the  points  (x^,  y^)  and  (ajg,  2/2)  are  on  the  same  side  or  oppo- 
site sides  of  the  line  Ax-^By-[-C  —  0  according  as  Axi-\-  Byi-\-C 
and  Ax2  +  By2  +  C  have  the  same  sign  or  opposite  signs.  This 
proves  for  the  straight  line  the  principles  illustrated  in  §§  14-20. 

48.  To  find  the  equations  of  the  bisectors  of  the  angles  bettveen  the 
lines 

Ax -\-By+C=0,     or      x cos  a  +  y  sin  «  — p  =  0,  (1) 

and         A'x-\-B'y-\-  O  =  0,     or   xcos  a' -^-ysina' —2)'=0.  (2) 

Suppose  the  equations  of  the  lines  written  so  that  the  origin  is  on 
the  same  side  of  both  lines. 

Then  for  any  point  (x,  y)  on  the  bisector  of  the  angle  which 
includes  the  origin, 

Dist.  from  (1)  to  (x,  y)  =  Dist.  from  (2)  to  (x^  y)  ; 

and  for  any  point  (a;,  y)  on  the  other  bisector, 

Dist.  from  (1)  to  {x,  y)  =  —  Dist.  from  (2)  to  (x,  y). 

Therefore  the  required  equations  are  [§  47] 

V^2  +  £2  V^/2  +  B/2    •  ^  ^ 

or  a;cosa  + 2/sina-j»  =  ±  (ajcosa' 4- 2/sina' -^')»  (4) 

Ex.    Show  that  these  two  lines  are  perpendicular  to  each  other.     [Use  (4).]  • 


60  THE   STRAIGHT   LINE  [49 

EXAMPLES 
Find  the  following  distances  : 

1.  From  3 a; +  4?/ +  10  =  0  to  (1,  12),  (-3,  -9),  (3,4). 

2.  From  x-Sy  =  7  to  (3,  2),   (6,  3),  (2,  -5). 

3.  From  5a;  +  12y  =  13  to  (3,  -2),  (-3,2),   (4,  -7). 

4.  From  ?>(x  — «)+  a(y  —  b)  =0  to  (—a,  —  &),   (-  &,  —  «),  (&,  «). 

5.  From  4(x-3)=3(2/+l)  to  (6,1),  (4,  -5),  (-7,2). 
Are  the  above  points  on  the  same  or  opposite  sides  of  tlie  lines  ? 
Find  the  equations  of  the  bisectors  of  the  angles  between  the  lines 

6.  Sx  +  4y  +  12  =  0  and  4x-Sy  =  12. 

7.  3  a:  -  4  ?/  +  5  =  0  and  12  a;  +  5  «/  +  14  =  0. 

8.  2/  =  2x+5  and  x  —  2y  =  8. 

^  9.  y=V3x+3  and  ic  +  V3 y  =  9. 

,J\  10.  Find  the  lengths  of  the  altitudes  of  the  triangle  whose  vertices  are  (3,  4), 

(-4,1),  and  (-1,  -5). 

\    11.  What  is  the  locus  of  a  point  which  is  3  units  distant  from  the  line 
2x-42/  =  9? 

f     49.    To  find  the  equation  of  a  straight  line  passing  through  the  inter- 
section of  two  given  straight  lines. 

The  most  obvious  method  of  finding  the  required  equation  is  to 
find  the  coordinates  x\  y'  of  the  point  of  intersection  of  the  two 
given  lines,  and  then  substitute  these  values  in  equation  (3),  §  43. 

The  following  method  of  dealing  with  this  class  of  problems  is, 
however,  sometimes  preferable,  both  on  account  of  its  generality  and 
because  it  saves  the  labor  of  solving  the  two  given  equations : 

Let  the  equations  of  the  two  given  straight  lines  be 

Ax  +  By-^C=0,  (1) 

and  A'x  +  B'y  +  C"  =  0.  (2) 

The  required  equation  is  then  written 

Ax  +  By  +  C  +  \  (A'x  +  B'y  +  C)  =  0,  (3) 

where  A  is  any  constant. 


V^ 


49]  THE   STRAIGHT   LINE  61 

Equation  (3)  is  of  the  first  degree,  and  therefore  represents  a 
straight  line ;  if  (x',  y')  is  the  point  common  to  (1)  and  (2),  we  have 

Ax'  -j-By'+C  =  0 
and  A'x'-}-B'y'-{-C'  =  0. 

/.  Ax'  +  By'+C  +  X  (A'x'  +  B'y'  +  C)  =0,  -  - 

which  shows  that  the  point  (x',  y')  is  also  on  (3). 

Hence  (3)  is  the  equation  of  a  straight  line  passing  through  the 
point  of  intersection  of  the  two  given  lines.  Moreover,  equation  (3) 
contains  one  arbitrary  parameter,  A,  and  therefore,  by  giving  a  suit- 
able value  to  A.,  the  line  may  be  made  to  satisfy  any  other  given  con- 
dition; it  may,  for  example,  be  made  to  pass  through  any  other  given 
point,  may  be  made  parallel,  or  perpendicular  to  a  given  line.  Hence 
equation  (3)  represents,  for  different  values  of  A.,  all  straight  lines 
through  the  point  of  intersection  of  (1)  and  (2). 

The  other  condition  which  any  particular  line  is  made  to  satisfy 
will  give  an  equation  for  the  determination  of  the  value  of  \. 

Ex.  Find  the  equation  of  a  straight  line  passing  through  the  point  of  inter- 
section of2a;4-52/-4  =  0  and  4x  —  2?/+2  =  0,  and  perpendicular  to  the  line 

2x-4ij  =  7.  (1) 

Any  line  through  the  intersection  is  given  by 

2x-\-5y-4  +  \(Ax-2y-\-2)=0, 
or  (2  +  4  \)x  +  (5  -  2  \)  ?/  +  (2  X  -  4)  =  0.  (2) 

Now  (2)  is  perpendicular  to  (1)  if  (§  45,  III) 

2(2  +  4  X)  -  4(5  -  2  X)  =  0  ;  i.e.  if  X  =  1. 
.'.  6  X  +  S  y  =  2m  the  required  equation. 

EXAMPLES 

''^     1.   Find  the  equations  of  the  lines  joining  the  points  (0,0),  (4,  2),  (—  1,  3), 
(_  3^  —  4)  to  the  point  of  intersection  of  the  lines  2x  +  y  =  2  and  2  ic  —  3y  =  6. 
2.    What  is  the  equation  of  the  straight  line  passing  through  the  intersection 
oiix-2y  =  4:  and  7  a;  -  3  y  +  21  =  0,  and  parallel  to9x-iy  =  0? 

/^  3.  Find  the  equations  of  the  two  lines  passing  through  the  intersection  of 
t  —  2y  —  \  and  2x  |-5?/  +  4  =  0,  the  one  parallel,  the  other  perpendicular,  to 
z  -f-  2  j^  =  0. 

4.  Find  the  equations  of  the  two  lines  passing  through  the  intersection  of 
7  x  —  5  ?/  =  35  and  8  a;  —  3  y  +  24  =  0,  the  one  parallel  to  y  =  2  x,  the  other  per- 
pendicular to  3  a;  +  4  2/  =  0. 


62  THE   STRAIGHT   LINE  [50 

^  6.  Show  that  it  S  =  0  and  S'  =  0  represent  the  equations  of  any  two  loci  with 
terms  all  transposed  to  the  first  member,  and  \  denotes  an  arbitrary  constant, 
then  the  locus  represented  by  the  equation 

will  pass  through  all  the  common  points  of  the  two  given  loci. 
Consider  the  two  cases  X  =  0  and  \  =  co . 

6.   Find  the  equation  of  the  circle  which  passes  through  the  origin  and  the 
common  points  of  the  circles 

x2  +  ?/2  =  25  and  a;2  +  2/2  _  18a:  +  20  =  0. 

V      7.   Find  the  equation  of  the  circle  which  passes  through  the  common  points  of 

a;2  4-  ?/2  =  16  and  x  —  y  =  4, 

and  (1)  passes  through  the  origin,  (2)  touches  the  aj-axis. 

50.    To  Jind  the  equation  of  a  straight  line  referred  to  axes  inclined 
at  an  angle  <u. 

Y 


Let  ABP  be  any  line  meeting  the  y-axis  at  a  distance  h  from  the 
origin,  and.  making  an  angle  y  with  the  cc-axis. 

Draw  PQ  parallel  to  the  ?/-axis  and  OR  parallel  to  the  given  line 
ABP. 

Let  P(x,  y)  be  any  point  on  the  line  ABP.,  then 

OQ  =  x,  and  QR=  QP- RP  =  y-b. 

.  Since  Z  Oi?Q  =  Z  i20  F=  o>  —  y,  we  also  have 


51]  THE   STRAIGHT   LINE  63 

y —  b  _QIi  _  sin  QOE  _       sin  y 

X         OQ     sin  ORQ     sin  (<o  —  y) 

,.y=    .    f  y       x  +  b,  (1) 

sin  (ci>  —  y) 

which  is  the  required  equation.  * 

Let  m=  .  ^^y      =-. ^-^^ (2) 

Sin  (o>  —  y)      sin  (t)  —  cos  o>  tan  y 

mt-  J.  *W  sin «  /ON 

Then  tan  v  =  ,— ,  (3) 

'     1  +  wt  cos »'  ^  ^ 

and  equation  (1)  becomes         y  =  ma?  +  b,  (4) 

which  in  oblique  coordinates  represents  a  straight  line  inclined  to 

the  a?-axis  at  an  angle  tan-^f.,  ^^^^°"     ). 
^  \l  +  »*cos«/ 

51.  Some  of  the  investigations  in  the  preceding  sections  of  this 
chapter  apply  to  oblique  as  well  as  to  rectangular  axes.  Let  the 
student  show  that  this  is  true  of  the  following  equations : 

-  +  f=l.  [(!),§  39.] 

CL        0 


y-y,  =  m{x-x,\  [(3),  §  43.] 

[(3),  §  44.] 


EXAMPLES  ON  CHAPTER  III 

1.    What  are  the  loci  of  the  following  equations  ? 

(1)   x'^  +  axy  =  {i.  (2)   x^-xy'^z=iO. 

(3)   a;3  +  ?/8  =  o.  (4)   x^-y^=Q. 

(5)   a2^,2_  52^2  =  0.  (6)   a2a;2^.^,2y2  =  o. 

(7)    (a;2-l)(y2_4)^o.  (8)    (ax  +  6y)2  =  A 

(9)   2/2-(a;-a)2  =  0.  (10)    (x- a)2+ (y- 6)2  =0. 

(11)    (x  -  a)2  -  (// -  6)2  =  0.  (12)   x3-x2y  +  xy2_y2  =  o. 

(13)   /)  =  asec  (^-a). 


64  THE   STRAIGHT   LINE  [51 

2.    Find  the  equations  of  the  lines  which  bisect  the  opposite  sides  of  the 
quadrilateral  (3,  4),  (5,  1),  (-  3,  4),  and  (5,  -  1). 

■^  3.    Find  the  equations  of  the  lines  which  go  through  the  origin  and  trisect 
that  portion  of  the  line  Sx  —  2y  =  IS  which  is  intercepted  between  the  axes. 

4.  Find  the  equation  of  the  line  through  (a,  b)  parallel  to  the  line  joining 
(0,  -a)  and  (fc,  0). 

5.  Find  the  equations  of  the  lines  which  pass  through  (—2,  1)  and  cut  off 
equal  lengths  from  the  axes. 

^    6.   Show  that  the  three  lines  2x  —  y  =  4:,  x-{-2y  =  7,  and  Sx  +  y  =  11  meet 
in  a  point, 

7.    Show  that  the  three  points  (1,  3),  (—1,  4),  and  (9,  —  1)  are  on  a  straight 
line  ;  also  (3  a,  0),  (0,  3  6),  and  (a,  2  b). 

/     8.    For  what  value  of  m  will  the  line  y  =  mx  —  4  pass  through  (4,  2)  ?    be  2 
units  distant  from  the  origin  ? 

9.    A  line  is  3  units  distant  from  0  and  makes  an  angle  of  60°  with  OX. 
What  is  its  polar  equation  ?   its  rectangular  equation  ? 

10.   Find  the  locus  of  all  points  which  are  equidistant  from  the  two  lines 
Sx-2y  =  S  and  Sx-2y  +  2  =  0. 
}/       11.    What  is  the  distance  between  the  parallel  lines 

3 X  +  4 ?/  =  5  and  6x-\-Sy-\-lb  =  0? 
12.   Find  the  points  on  the  axes  which  are  4  units  from  the  line 
x-7y-h21=0. 

^  13.  Show  that  the  perpendiculars  let  fall  from  any  point  of  22x  —  4y  =  15 
upon  the  lines  24:X-\-T  y  =  20  and  4x  —  Sy  =  2  are  equal.  Find  another  line 
of  which  this  statement  is  true. 

14.  Find  the  perpendicular  distance  of  the  point  (I,  m)  from  the  line  through 
(a,  b)  perpendicular  iolx-{-  my  =  \. 

*  16.  Show  that  the  bisectors  of  the  interior  angles  of  a  triangle  meet  in  a 
point. 

16.   Find  the  locus  of  a  point  which  is  equally  distant  from  the  lines 

5 aj  -  3  ?/  =  15  and  Zy  =  5x^Q. 

»        17.   Show,  by  the  use  of  (1),  §  42,  or  by  transforming  (3),  §  43,  that  the  polar 
equation  of  a  line  passing  through  the  fixed  point  (pi,  ^i)  may  be  written 
p  cos  {d  —  a)  =  p\  cos  (^1  —  a). 

18.  Show,  directly  from  a  figure,  or  by  transforming  (3),  §  44,  that  the 
polar  equation  of  the  straight  line  which  passes  through  the  two  fixed  points 
Cpi,  ^i)  and  (/02,  ^2)  is 

pipo  sin  (02  —  61)  +  P2P  sin  (d  —  62)  +  ppi  sin  (^1  —  6)  =0, 


51]  THE  STRAIGHT  LINE  66 

^    19.    Show  that  the  equations 

n 
^  COS  ^  +  -B  sin  0  +  -  =  0,     ^  cot  ^  =  IT,   V 

P 
p  =  kaec{d  —  a),  p  =  icsc  (^- /3), 

represent  straight  lines. 

20.  Show  that  the  equations  of  the  lines  passing  through  (—3,  2)  and 
inclined  at  an  angle  of  60°  to  the  line  VSy  —  x  =  'S  are 

ic  +  3  =  0  and  VSy  +  x  +  3  =  2  V3. 

/       21.    Find  the  equations  of  the  sides  of  a  square  of  which  the  points  (2,  2) 
and  (—2,  1)  are  opposite  vertices. 

22.  What  are  the  equations  of  the  sides  of  a  rhombus  if  two  opposite  vertices 
are  at  the  points  (—  1,  3)  and  (5,  —  3),  and  the  interior  angles  at  these  vertices 

/e  each  60°  ? 
23.    Prove  that  the  equation  of  the  straight  line  which  passes  through  the  point 
(a  cos^  6,  a  sin^  d)  and  is  perpendicular  to  the  straight  line  x  sec  6  -{-  y  esc  ^  =  a  is 

X  cos  6  —  y  sin  6  =  a  cos  2  0. 

24.  Show  that  the  equations  of  the  lines  passing  through  the  point  (4,  4) 
and  whose  distance  from  the  origin  is  2  are  x{l  ±  y/1)  +  1/(1  T  V^)  =  ^' 

25.  Find  the  area  of  the  triangle  formed  by  the  lines 

y  +  3a;  =  6,   y  =  2a:-4,   ?/  =  4x  +  3. 

26.  Show  that  the  area  of  the  triangle  formed  by  the  linea 

y  =  m\X  +  6i,   y  =  m^x  +  62,  and  «  =  0, 

is  1   (&1  -   &2)^ 

2  m\  —  m2 

27.  Show  that  the  area  of  the  triangle  formed  by  the  lines 

y  =  mix  +  61,   y  =  m^x  +  ^2,   and  y  —  mspc  -f  63 

is  1  r(ft.-W  +  (6.-W'  +  (fes-ft.)'] .  (Use  Ex.  26.) 

2  L  wii  —  m2        m2  —  ms        ma  —  mi  J 

28.  What  is  the  equation  of  a  line  passing  through  the  intersection  of 
3a;  —  2?/  + 12  =  0  and  x  +  4  y  =  20,  and  (a)  equally  inclined  to  the  axes  ? 
(6)  whose  slope  is  —  2  ? 

29.  The  distance  of  a  line  from  the  origin  is  6,  and  it  passes  through  the 
intersection  of  2x  +  Sy  =  6  and  3a;  —  6y  +  29  =  0.     Find  its  equation. 

30.  Find  the  equations  of  the  two  lines  which  pass  through  the  intersection 
of  a;  +  2 y  =  0  and  2x  —  y  +  S  =  0,  and  touch  the  circle 

x2  +  yi  =  9. 


ee  THE   STRAIGHT  LINE  [51 

31.  Find  the  equations  of  the  two  lines  which  pass  through  the  intersection 
of  x  +  Sy  +  9  =  0  and  3 x  =  ?/  +  13,  and  touch  the  circle 

(x  +  2)2  +(y-  3)2  =  25. 

32.  Find  the  equations  of  the  diagonals  of  the  rectangle  whose  sides  are 
x-\-2y  =  10,  x  +  2y  +  2  =  0,  2x-  y  =  12,  and  2x  —  y  =  lQ,  without  finding 
the  coordinates  of  its  vertices. 

33.  A  circle  passes  through  the  common  points  of 

a;2  +  2/2  _  25  and  a;  -  4  y  +  13  =  0, 
and  cuts  the  avaxis  in  two  coincident  points.    Find  its  equation. 

34.  Show  that  the  locus  of  a  point  which  moves  so  that  the  sum  of  its  dis- 
tances from  the  two  lines 

X  cos  a  -{-  ysin  a=p  and  x  cos  cc'  +  y  sin  a'  =  p' 
is  constant  and  equal  to  K  is  the  straight  line 

2  X  cos  ^  (a  +  a')  +2y  sin  H«  +  a')  =  (p  -\- p'  -\-  K)  sec  \  (a  -  a'). 

Show  that  the  locus  is  parallel  to  one  of  the  bisectors  of  the  angles  formed  by 
the  two  given  lines. 

Show  also  that  if  the  difference  of  the  distances  from  the  two  given  lines  is 
constant,  the  locus  is  a  straight  line  parallel  to  the  other  bisector. 

35.  If  p  and  p'  be  the  perpendiculars  from  the  origin  upon  the  straight  lines 
whose  equations  are 

X  sec  6  -\-  y  esc  0  =  a  and  x  cos  6  —  y  sind  =  a  cos  2  0, 
prove  that  4p^  +  p'^  =  a\ 

36.  Show  that  the  equation  of  the  line  passing  through  the  points  (a  cos  a, 
6sin«)  and  (acosjS,  6sin)3)  is 

bx  cos  ^  (a  +  ^)  +ay  sin  ^  (a  +  j3)  =  a&  cos  ^  (a  —  p). 

37.  Show  that  the  equation  of  the  line  which  passes  through  the  points 
(a  sec  a,  b  tan  a)  and  (a  sec  /3,  b  tan  /3)  is 

bx  cos  I  (a  —  /3)  —ay  sin  ^  (a  +  j3)  =  a&  cos  |  («  +  /3). 

38.  Show  that  the  three  straight  lines 

aix  +  biy  +  ci  =  0,   azx  +  b2y  +  C2  =  0,   a^x  -\-  bsy  +  cs  =  0 

will  meet  in  a  point  if 

ai,   &i,   Ci 

a2,  62,  C2   =  0. 

39.  Find  the  determinant  expressions  for  the  coordinates  of  the  vertices, 
and  for  the  area  of  the  triangle  formed  by  the  three  lines  in  Ex.  38,  and  show 
that  the  determinant  there  given  is  a  square  factor  of  the  determinant  expression 
for  the  area  of  the  triangle. 


y 


CHAPTER   IV 


TRANSFORMATION    OF   COORDINATES,    OR    CHANGE    OF  AXES 


52.  The  formulae  for  changing  an  equation  from  rectangular  to 
polar  coordinates  and  vice  versa  have  already  been  found  in  §  6,  and 
their  usefulness  amply  illustrated.  Moreover,  the  equation  of  a 
curve  in  any  system  of  coordinates  is  sometimes  greatly  simplified 
by  referring  it  to  a  new  set  of  axes  of  the  same  system.  Hence,  it  is 
also  desirable  to  be  able  to  deduce  from  the  equation  of  a  curve 
referred  to  one  set  of  axes  its  equation  referred  to  another  set  of 
axes  of  the  same  system.  Either  of  these  operations  is  known  as  a 
Transformation  of  Coordinates,  or  Change  of  Axes. 

The  equations,  which  express  the  relations  between  the  two  sets 
of  coordinates  of  the  same  point,  and  by  means  of  which  these  opera- 
tions are  performed,  are  called  Formulae  of  Transformation. 

53.  To  move  the  origin  to  the  point  (7i,  k)  without  changing  the  direc- 
tion of  the  axes. 

Let  OX  and  0  F  be  any  pair  of  axes  inclined  at  an  angle  w,  and 
let  O'X'  and  O'Y'  be  a  new  pair  parallel  respectively  to  the  old. 
Let/  P  be  any  point  whose  coordinates  are  (x,  y)  with  respect  to  the 
original  axes,  and  (x\  y')  with 
respect  to  the  new  axes. 

Then  from  the  figure, 

OQ=ON-^NQ, 
and  QP=QE-^EP. 

But  OQ=x,  NQ=x',  ON=^h, 
QP=y,  RP=y',  QR^k. 

.'.  oc  =  gc'  -\-h9 


y  =  y'^k. 


(1) 


68 


CHANGE   OF   AXES 


[54 


As  these  equations  are  independent  of  w,  they  hold  for  both  rec- 
tangular and  oblique  coordinates. 

Hence  to  find  what  a  given  equation  becomes  when  the  origin  is 
moved  to  the  point  (h,  k),  the  new  axes  being  parallel  to  the  old, 
substitute  x'  +  h  for  x  and  y'  -\-k  for  y.  After  the  substitution  is 
made  we  can  write  x  and  y  instead  of  x'  and  y' ;  so  that  practically 
this  transformation  is  effected  by  simply  writing  x-{-h  in  the  place 
of  X,  and  y  -{-k  in  the  place  of  y. 

54.  To  transform  from  one  set  of  rectangular  axes  to  another,  having 
the  same  origin. 

Let  {x,  y)  be  the  coordinates  of  any  point  P  referred  to  the  old 
axes  OX  and  OF;  and  {x\  y')  the  coordinates  of  the  same  jDoint 
referred  to  the  new  axes  OX'  and  OY'.     Let  the  angle  XOX '  =  0. 

Draw  the  ordinates  MP  and  J^P,  and  the  lines  QJ^  and  BN  par- 
allel to  OX  and  OY  respectively. 

Then        Z  NPQ  =  6, 

OM=x,  MP^y, 

ON=x',  NP=7j\ 

OR=ONGose  =  x'eose, 

BN=  ONsin  e  =  x'  sine, 

QN=  NP sine  =  y'  sinOj 

QP  =  NP  cos  e=y'  cos  9. 

But       OM=OR-QN, 

and  MP  =  EN-j-  QP. 


Therefore 
and 


a?  =  a5'cos6  -  2/'sm0, 1 
2/  =  iK'  sin  6  +  y'  cos  B.  j 


(1) 


If  at  the  same  time  the  origin  he  changed  to  the  ptoint  (h,  k),  the  re- 
quired formulm  will  he 


a?  =  a?'  cos  6  -  y'  sin  6  +  ^, 
2/  =:  a?'  sin  e  +  y'  cos  0  + 


Tc,] 


(2) 


This  transformation  is  clearly  obtained  by  combining  the  two 
formulae  (1)  and  (1)  of  §  53. 


54]  CHANGE  OF   AXES  69 

EXAMPLES 

Transform  to  parallel  axes  through  the  point  (3,  —  2) 
1.    2/2_4a; +4y  +  16  =  0.  2.    2x^  +  3y^  -  12  x  +  12y +  29  =  0. 

What  are  the  equations  of  the  following  loci  when  referred  to  parallel  axes 
through  the  point  (a,  6)  ? 

3.    (x  -  a)2  +  (y  -  6)2  =z  r^.  4.   xy  -  bx  -  ay  -\-  ah  =  a^. 

6.  1/2  _  2  6y  +  4  ax  =  4  a2  _  62.  e.   x^  -  y2  _  2  ax  +  2  62/  =  6^  -  a^. 

7.  62(aj2  _  2  ax)  +  a2(?/2  _  2  &«/)  +  a2&2  =  0. 

Transform  by  turning  rectangular  axes  through  an  angle  of  45°. 

8.  x2  -  2/2  =  flj2.  9.   3x2  -  2 xy  +  3 2/2  =  32.     (10.  p.  45.) 
10.   2(y  +  x)  =  (y-  x)K                  11.   ax2  +  2  tey  +  ay^  =  1. 

12.   x^  +  2/^  =  ai  13.    (x^  +  y'^y  =  4a^xY- 

In  13  change  the  given  equation  and  the  result  to  polar  coordinates. 


yl!l4.   Transform  -  4-  ^  =  1  by  turning  the  axes  through  tan-i^  * 
a     b  b 

"f-  15.  What  does  2x2  —  3x2/ —  2^/2  —  50^2  become  when  the  axes  are  turned 
through  tan-i  —  2  ? 

16.    If  the  axes  be  turned  through  an  angle  of  30°,  what  does  the  equation 
9x2  -  2  V^xy  +  112/2  =  4  become  ? 

/  17.  Show  that  the  equation  2x2  4-x2/  —  2/^  +  5x  —  2/  +  2  =  0  can  be  reduced 
to  2  x2  4-  xy  —  y^  =  0,  by  transforming  to  parallel  axes  through  a  properly  chosen 
point. 

Through  what  angle  must  the  axes  be  turned  to  cause  the  term  in  xy  to  disap- 
pear from  the  following  equations  ? 

18.   x2-6xy  +  y2  =  i6.  19.    8x2  +  4xy  +  5y2  =  36. 

20.    (4  2/  -  3  x)2  -  20  X  +  110  2/  =  75.  21.    ax2  -{.2hxy  +  by^  =  c. 

22.    Show  that  the  transformation  a:  =  ^,   2/  =  x  simply  changes  the  scale  of 

the  curve,  k  being  the  factor  of  magnification. 

28.   Compare  the  curves  y  =  sin  x  and  2/  =  i  sin  2  x. 

24.    Show  that  the  curve  y  =  sin2  x  differs  only  in  position  and  size  from 
y  =  sin  X. 


CHAPTER   V 
SLOPE,  TANGENTS,   AND  NORMALS 

55.  It  sometimes  happens,  that  the  substitution  of  a  particular 
value  for  the  variable  in  a  fraction  causes  both  numerator  and  de- 
nominator to  vanish,  and  the  fraction  takes  the  form  ^. 

Thus,  z —  becomes  j-  when  x  =  ^. 

-   .  1  —  CSC  a;  0  2 

The  fraction  is  then  said  to  be  indeterminate  ;  that  is,  the  fraction 
has  no  value,  or  meaning,  for  this  particular  value  of  the  variable. 
Such  a  fraction,  however,  usually  approaches  a  defiyiite  limit  as  the 
variable  approaches  this  particular  value  as  its  limit.  This  limit  is 
the  value  we  then  assign  to  the  fraction,  because  it  fits  in  continu- 
ously with  the  other  values  of  the  fraction.  This  definite  limit  can 
be  found  by  reducing  the  given  fraction  to  an  equivalent  one  whose 
terms  do  not  both  vanish  when  the  particular  value  is  substituted 
for  the  variable. 

In  all  the  investigations  which  follow  in  this  chapter  it  will  be 
found  to  be  necessary  to  determine  the  limit  which  a  ratio  approaches 
when  its  terms  both  approach  zero.  Hence  the  student  should  now 
fix  in  mind  the  following  definition,  viz . : 

A  constant  is  called  the  limit  of  a  variable  if  the  difference  between 
the  constant  and  the  variable  can  be  made  to  become  and  remain  as 
small  as  ive  please. 

56.  Examples  of  limiting  values  of  ratios. 
(1)  Let  Khe  the  area  of  a  square  whose  side  is  x. 

Then  rlil^L^l      =  5.    But  ^'^^  ^  =  i^"^.  ^  =  Jl%  (x)  =  0. 

*  The  sign  "  =  "  in  these  conditions  for  a  limit  should  be  read  "  approaches." 

70 


56]  SLOPE,   TANGENTS,   AND  NORMALS  71 

(2)  Let  K  be  the  area  of  a  rectangle  with  a  constant  base  h  and  a  variab)e 
altitude  x. 

Then  pill^l      =5.     But   ^^!"  :^=1'[^^=:6. 

LliraxJ;,^o     0  ^=^  X      ^=^  X 

(3)  Let  V  be  the  volume,  T  the  total  surface,  C  the  circumference  of  the 
base  of  a  right  circular  cylinder  whose  altitude  is  constant  and  radius  variable. 

Then  r^^^^l        =^'     \^^^^JL^        =2. 

LlimCJr=o     0      LlimFJr=o     0 

,  lim    T  _   lim   2  7rr(r  +  h)       lim    2  (r  + /i)      2 /i 

If  S  be  the  convex  surface,  find     ^!^^  — • 

Llim  (x2  -  a-2)  J^^«     0' 

lim    (x  —  a)2 _    lim    a;  —  «_a 

^-(^x'-a^~^  =  o,x  +  a~   ' 


Multiplying  both  numerator  and  denominator  by  1  +  y/l  —  x^  gives 

lim    1  -  Vl  -  a;-'  _    lim   x^ lim  1  _1 

x  =  0         ^2  -x  =  0^,^^_^^.^--^^-x  =  0Y:^-/==-1' 

EXAMPLES 
Find  the  limits  indicated  in  the  following  expressions  : 
-       lim    x^-gg  „      lim    x^  -  a^  ,     lim  (x-a) 


x  =  ay^2_a^  x  =  ay;2_a2  '  x-ax^-ax'^-a'^x  +  a^ 

^      lim    3  a;2  -  6  a;  +  3        _      lim  x'^  lim    2  x^^  +  a;  -  1 

•  x  =  l2x2-4x  +  2'         '  ^-^a-VS^^T^^'         **•  x  =  co    x^^x  +  2' 


lim    V4+X-V4-X  lim      /  ,  ,     ,  9      Hm   sin^.j 

,Q       lim     sec  x  _  ■.  ,,       lim    1  —  cos  x  _  1         -g      li"i   tan  x  — sin  x_/^ 

'  x-^^°t'dnx~   '  '  x  =  0    sin2x     ~2*  *  x  =  0    i_cosx   ~  ' 

,0      lim  sinx  _   lim   tanx  _  -,  ^m      lim  sec  x  —  1  _  1 

***•  X  =  0    a;    ~  X  =  0     y,     ~    •  ■^*'  X  =  0       a;2       ~  2* 

16.  If  V  be  the  volume,  T  the  total  surface,  ^S*  the  convex  surface,  C  the 
circumference  of  the  base  of  a  cone  of  revolution  whose  altitude  h  is  constant, 

oi,«™  *u»*      lim    T     h       lim    T  lim    T     .         lim     T     « 

show  that        .^_  =  -,        .^  —  =  Go,         .^  —  =1,         .       —  =  2. 


72 


SLOPE,   TANGENTS,   AND   NORMALS 


[57 


57.  Definitions.     Let  two  points  P  and   Q  be  taken  on  any 
curve  PQMf  and  let  the  point  Q  move  along  the  curve  nearer  and 

nearer  to  P;  the  limiting  position,  TT', 
of  the  secant  PQ  when  the  point  Q  ap- 
proaches indefinitely  near  to  P  is  called 
the  Tangent  to  the  curve  at  the  point  P. 
The  straight  line  PN  through  the  point 
P,  perpendicular  to  the  tangent  TT',  is 
called  the  Normal  to  the  curve  at  the 
point  P. 

The  Slope,  or  Gradient,  of  a  curve  at  any 

point  is  the  slope  of  the  straight  line  tangent  to  the  curve  at  that 

point. 

58.  To  find  the  slope  of  a  curve  at  any  point* 

Y 


Let  P(x,y)  and  Q(x  +  Sx,  y -h  ^y)  be  two  points  close  together 
on  any  curve  AB ;  then  8x  is  the  difference  of  the  abscissas,  Sy  the 
difference  of  the  ordinates  of  P  and  Q. 

Let  the  secant  PQ  meet  the  ic-axis  in  S,  and  let  the  tangent  line 
at  P  meet  the  a>axis  in  T. 

Draw  the  ordinates  MP,  NQ,  and  draw  PR  parallel  to  the  a;-axis. 

Then  PR  =  hx,     RQ  =  Sy. 

Let  the  equation  of  the  curve  be 

y^m-  (1) 


Bead  Ex.  1,  §  59,  in  connection  with  this  general  demonstration. 


58]  SLOPE,   TANGENTS,   AND  NORMALS  73 

Then  at  the  points  P  and  Q  we  have 

OM=x,  MF  =  y=f(x), 

ON=x  +  ^,     NQ  =  y  +  8y=f(x-\-8x). 

.'.8y=:f(x  +  Sx)-f(x).  _  _ 

Also  tan  XSQ  =  tan  RPQ  =  ^  =  ^ . 

PR     &c 

.•.tanX^Q  =  |^==£(^±MziiM.  (2) 

oX  ox 

The  slope  of  the  tangent  TP,  which  is  the  slope  of  the  curve  at 
the  point  P,  is  the  ultimate  slope  of  the  secant  SPQ  when  the  point 
Q  moves  along  the  curve  close  up  to  P;  i.e. 

tan  XTP=  lim  tan  XSQ  =  lim  -^  as  Q  approaches  P. 

ox 

When  the  point  Q  approaches  the  position  of  P  as  a  limit,  the  dif- 
ferences Bx  and  8y  simultaneously  approach  zero  as  a  limit,  and  the 

limiting  value  of  the  ratio  -^  is  denoted  by  -^  ;  therefore  in  the  limit 

ox  ax 

we  have 

The  ratio  represented  by  the  last  member  of  equation  (3)  is  also 
a  function  of  x ;  and  if,  x  being  regarded  as  fixed,  this  ratio  has  a 
definite  limiting  value  as  Sx  approaches  zero,  this  limiting  value  is 
called  the  Derived  Function,  or  the  Derivative  of  f(x)  with  respect  to 
X,  and  will  be  denoted  hy  f'(x),  or  i)x[/(^)]  > 

i.e.  if  V=f{ic),  then  ^|  =f\x)  =  I)^if{x)^. 

Hence  to  find  the  slope  at  any  point  of  a  curve  whose  equation  is 
in  the  form  y  =f(x)  we  find/'(x),  the  derivative  oi  f(x)  with  respect 
to  Xf  and  in  this  substitute  the  abscissa  of  the  given  point. 

To  find  the  derivative  of  a  function  of  a;,  denoted  by  /(»),  we 
assign  a  small  increment  Sx  to  ic,  producing  an  increment,  denoted 
by  f(x  -{-  8x)  —/(«),  in  the  function,  and  then  find  the  limiting  value 
of  the  ratio 

Sx 


74 


SLOPE,  TANGENTS,   AND  NORMALS 


[59 


59.   Examples  of  derivatives  and  slope  of  curves. 
Ex.  1.  Find  the  slope  of  the  curve  whose  equation  is 

y  z=x'^  -\-  a. 


(1) 


Let  P{x,  y)  and  Q{x  +  5a;,  y  +  8y)  be  any 
two  points  close  together  on  the  curve ;  and 
let  TP  be  the  tangent  at  P. 

Then  at  P,        y  =  x^  -\-  a,  (2) 

and  at  §,      y  -\-  Sy  =  (x  +  Sxy  +  a.  (3) 

Whence 

(y  +  5y)-y^(x-j-  8xy  +  a  -  (a;^  +  a) 
5x  5x 

=  tan  EPQ.  (4) 

.:^  =  2x-{-8x  =  tan  XSQ.  (5) 

dx 

When  Q  approaches  P,  or  as  we  say,  pro- 
ceeding to  the  limit  5x  =  0,  we  have  (§  58) 


dy  ^ 
doc 


J?  X  =  tan  XTP, 


(6) 


Hence  the  slope  of  the  curve  at  any  point  is  equal  to  twice  the  abscissa  of  the 
point. 

At  Po,     x  =  Q. 

.'.  PqTq  is  parallel  to  the  x-axis. 
At  Pi,     x  =  l, 

.-.  tan  XTiPi  =  1. 
At  Pa,     x  =  l 

.:  tan  XP2P2  =  3. 
At  P3,     X  =  —  ^, 

.-.  tan  XT3P3  =  -  1. 

Ex.  2.   Let  the  equation  of  the  (jiven  curve  he  y 
In  this  example  we  have  given /(cc)  =  x5.    Then  from  the  definition  of  the 
derivative  given  in  equation  (3)  of  §  58,  we  have, 

fir^\-    ^i^"    /(x  +  8x)-f(x)  _    lim    (x  +  8xy  —  x^ 
J  '-''>- 8x  =  0  ^  -Sx  =  0  sx 

=  3^*2  0  (^  **  +  10  xHx  +  10  xHx"^  +  5  x8x^  +  8x^)  =  5  a*. 

That  is,  the  slope  of  this  curve  at  any  point  (x,  y)  on  the  curve  is  equal  to 
five  times  the  fourth  power  of  the  abscissa  of  the  point. 


59]  SLOPE,   TANGENTS,   AND   NORMALS  76 

Ex.  3.    Find  the  slope  of  the  curve  y  =  — 

^  1 

We  now  have  from  the  definition  of  a  derivative,  since  f(x)  =-, 

z 

1  1 

dy  _    lim    x -\-  bx 
dx~  " 


lim 
5a;  =  0 

lim 

a;  +  5a;     x 
8x 

-1 

lim 
=  5x  =  0 

1 

a;  -  (a;  +  5a;) 
'  x(x  +  5a;)5x 

5a;  =  0 

a;(a;  +  5x) 

That  is,  the  slope  is  always  negative  and  varies  inversely  as  the  square  of 
the  abscissa  of  the  point. 

Ex.  4.   Let  y  =  Vx  be  the  given  curve. 

Then,  since  f(x)  =  Vx,  we  have  from  the  definition 

dy  _    lim    f(x  +  5x)  -  /(x)  _     lim     y/x  +  5x  —  Vx 
^^~5x  =  0  5a.  -5x  =  0  5a;  ' 

lim  1  ^1 

^^-^v^T5x  +  VS     2V^ 
Ex.  5.    To  find  the  derivatives  of  sin  x  and  cos  x.     * 
Let  5x  =  h,  for  convenience,  then  will 

i>,(sin  X)  =  /f  (J  sin(x  +  ^).,-sinx  ^  ^iim^  r    ^  /    ^  |\  sii^i^j 

i.e.  Da.(sina?)  =cosa5;  (Ex.  13,  p.  71.) 

D.(cos x)  =  j^%  co8(x  +  A)-cosx  ^  Mm ^  ^_  ^.^  /    ^  |\  sin|^j 

1.  e.  I>x  (cos  x)  =  —  sin  x. 

Check  the  results  found  in  Exs.  2,  3,  4,  and  5  by  constructing  the  loci. 

EXAMPLES 

Find  the  slope  at  the  points  where  x  =  0,  ±  1,  ±2,  etc.,  of  the  curves  whose 
equations  are  . 

/    1.   y  =  x\  *   2.   y  =  x4.  3.   x^  =  1.  4.   y^  =  x^ 

^'b.   y  =  x8-4x.  /  6.   y  =  x*-20x2  +  64.  ^7.   y  =  (^~p. 

X  —  £i 

8.   Find  the  slope  ot  y  =  Va^  +  x-^,  where  x  =  0,  ±  a,  oo  . 


9.   Find  the  slope  oi  y  =  Vd^  —  x^,  where  x  =  0,  ±  a,  ±  la. 
10.    Find  the  slope  of  10  ?/  =  x2  -  3  x  -  20,  where  x  =  0,  ±  1,  ±  4.    [§  22.] 


V 


y^ 


76  SLOPE,   TANGENTS,  AND  NORMALS  [60 

General  FoRMULiE  for  Differentiation 

60.    Tlie  derivative  of  the  product,  and  sum,  of  two  functions. 
Let  ff>{x)  and  F{x)  be  any  two  functions  of  x.     Then  (§  58) 

ox 

Introducing  <\>{x  +  8x)F(x)  —  <fi(x  +  8x)F(x)  in  the  numerator  gives 
D^[<t>{x)F{x)2  = 

i.e.         I>«,[<t»(a5)l^(a5)]  =  ^ix)F'(ix)  +  F(x)^>(ie).    [(3),  §  58.]       (3) 
By  an  extension  of  this  process  it  can  be  shown  that 

I>x[«|>i(a5)<|>2(a5)<|>3(a5) ...]  =  <t>i'(a5)<|>2(a5)<|)3(cc)  ...  +  4>2'(iK)<|>i(a5)«|>3(a?) ... 
+  <l>3'(a?)<|>i(a5)«|>2(a5) ...  +  ....  (4) 

Or,  as  a  special  case  of  (4),  we  have,  if  n  is  a  positive  integer, 

ALX«)]"  =  D,[<li(x)<l>(x)(fi(x)  -'ton  factors]  (5) 

=  [<A(^)]""V(a^)  +  [</>(aj)]""V(^)  +  •••  to  n  terms]     (6) 

E.g.  i>x(sin  x)^  =  S  (sin  xyD^isin  x)  =3  sin^  x  cos  a;.     [Ex.  5,  p.  75.] 

One  of  the  most  important  results  that  follows  from  (7)  is 

I>a,(cc")  =  nx^-^.  (8) 

In  like  manner  it  can  easily  be  shown  that 

I>a.[cf(x)'\  =  cf'(x},  where  c  is  a  constant,  (9) 

and  I>a5[<t>i(a5)  +  <|>2(a5)  +  <|>3(a5)+  ...]  =«|>i'(a?)+  <t)2'(a?)  +  <l>3'(ic)+.—   (10) 

Hence,  if  f(x)  is  a  rational  and  integral  algebraic  function  of  x 
(§  63),  f'(x)  is  found  by  midtiplying  the  coefficient  of  each  term  by  the 
exponent  of  x  in  that  term  and  diminishing  each  exponent  by  unity. 

E.g.    D^[x*  -  2  x3  +  4  x2  -  3  cc  +  6]  =  4  ^3  -  6x2  +  8 X  -  3. 


61]  SLOPE,   TANGENTS,   AND  NORMALS  77 

61.    To  find  the  derivative  of  a  function  of  the  type  F(x,  y)  =  0. 

When  we  desire  to  differentiate  a  function  of  the  type  F(Xj  y)  =  0, 
we  may  try  first  to  solve  the  equation  with  respect  to  ?/,  so  as  to  put 
it  in  the  form  y  =  f(x)  ;  or  to  solve  with  respect  to  x,  so  as  to  bring 
it  to  the  form  x=f{y).  It  is  useful,  however,  to  have  a  rule  to  meet 
cases  when  this  process  would  be  inconvenient  or  impracticable.  It 
will  be  sufficient  for  the  purpose  of  this  book  to  illustrate  the  rule 
by  considering  the  general  equation  of  the  second  degree  (§  87). 

Let         F{x,  y)  =  ax'  +  2  hxy  +  hy^  +  2  gx-it-2  fy  +  c  =  0.  (1) 

Let  P{x,  ?/)  and  Q{x-{-hx,  y-{-hi)  be  two  points  close  together 
on  the  locus  of  (1)  ;  then  at  P  and  Q,  respectively, 

ax^-\-2hxy  +  hy^-\-2gx-{-2fy-\-c  =  0,  (2) 

a{x  +  hxY  +  2  h(x  +  8x)  (y  +  8y)  +  b(y  +  Syf 

-^2g{x-\-dx)-^2f(y-{.Sy)+c  =  0.          (3) 

Subtracting  (2)  from  (3)  gives 
a(2  xBx  +  Sx2)  +  2  h(ySx  +  xSy  +  SxSy) 

+  6(2  ySy  +  Bf)  +  2  g8x-{-2  fhy  =  0.  (4) 

Whence  ^^ -  - ^  aa^  +  2  %  +  2^  +  a3a^  +  /% 

wnence  ^^-      2  hx  +  2  by-\-2f+b8y -hhSx  ^^A 

In  the  limit  when  &c  and  Sy  approach  zero,  we  have 

dy__  ax^^hy  +  g  ,„. 

dic~     hic  +  hy+f'  ^^^ 

Now  apply  to  (1)  the  rule  deduced  in  §  60  and  differentiate  first 
with  respect  to  x  regarding  y  as  constant;  then  differentiate  with 
respect  to  y  regarding  x  as  constant.  Denoting  these  partial  deriva- 
tives respectively  by  FJ(xy  y)  and  FJix,  y),  we  thus  obtain 

FJ(x,  y)=2{ax  +  hy  +  g),  (7) 

and  FJ{x,  y)  =  2  (hx  +  by  +.0-  (8) 

,   dy_  _  F^'Juo^  V)  ^  _  ax-{  hy  +  g  .g. 

*   dx~      Fy'ix,  y)         hx  +  by^f'  ^  ^ 

It  can  be  proved  that  this  formula  (9)  expresses  the  rule  for  differ- 
entiating any  function  of  the  type  F{Xy  y)  =  0. 


78  SLOPE,   TANGENTS,   AND  NORMALS  [62 

Tangents  and  Normals 

62.    To  find  the  equations  of  the  tangent,  and  the  normal  at  any 

point  (x',  y')  of  a  curve. 

dv' 
For  the  tangent,  m  =  /-,.  (§  58.) 

aoc 

dx' 
For  the  normal,  m  =  —  t-,-  (§  57  and  §  45.) 

dy' 
The  primes  in    ~-^  denote  that  the  coordinates  x',  y'  of  the  point 

of  contact  are  to  be  substituted  in  the  derivative  of  the  equation. 

Since  both  lines  pass  through  the  point  (x',  y'),  the  equation 
of  the  tangent  is  (§  46) 

y-y'=^,(^-^')',  (1) 

and  the  equation  of  the  normal  is 

V-y'  =  -^,   {oc-oc').  (2) 

CoR.     If  the  axes  are  oblique, 

dy  sin  -y  zo  ^a  \ 

doc     sm  C«  -  7)  ^'      ^ 

Hence  equation  (1)  holds  also  for  oblique  axes.* 

EXAMPLES  ON   CHAPTER  V 

Find  the  equations  of  the  tangent  and  the  normal  to  the  curve. 
1.    x2  +  ?/2  Z3  25  at  (3,  4).  2.    x2  +  ?/2  =  169  at  (- 12,  5). 

3.    ?/2  =  8xat(2,  4),   (8,  8).  4.    6  ?/ +  a;2  =  0,  at  (6,   -6). 

5.    y  =  x^-A:X2X  (2,  0),  (-  1,  3).      6.    y^  =  a;2,  at  (-  8,  4). 

7.  x2  +  ?/2  —  4  aj  +  6  y  =  0  at  the  points  where  x  =  0. 

8.  x2  +  1/2  +  4  X  —  6  y  =  12  at  the  points  where  x  =  2,  x  =  —  6. 

9.  x2  4-  ^/2  —  8  X  —  4  ?/  +  15  =  0  at  the  points  where  x  =  3,  x  =  5. 
10.    x2  +  ?/2  _  16  X  -  8  2/  4-  55  =  0  at  the  points  where  x  =  3,  x  =  5. 

*  The  theory  of  this  chapter  proves  what  has  hitherto  been  assumed  (see  note  on 
logic  of  plotting,  §  21),  viz.,  that  loci  of  equations  are  usually  smooth  curves  without 
sudden  changes  in  slope  or  curvature.  For,  since  the  slope  of  a  curve  f{x,  ?/)  =  0  at 
any  point  (x,  y)  is  a  function  of  x  and  y,  a  small  change  in  x  and  y  will  ordinarily 
produce  only  a  small  change  in  the  slope. 


> 


Ans. 

x'      yi 

Ans. 

y'     X'       • 

Ans. 

3x       y   ^1 
2x'      2y' 

Ans. 

3  X     2y     , 

x'       2/' 

Ans. 

^    +    2^=1. 
2x'     2y> 

Ans. 

xx'  +  2/2/'  =  1. 

62]  SLOPE,  TANGENTS,   AND  NORMALS  79 

Find  the  equation  of  the  tangent  to  each  of  the  following  curves  at  the 
point  (x',  y') : 

11.  y  =  x^. 

^  12.  2/2  =  X. 

13.  y  =  x^. 

^^  >    14.  y^  =  xK 

15.   xy  =  l. 

^    16.   x2  + 2/2  =  1. 
17.    x2- 2/2  =  1. 
^     18.    x^  +  y^  =  1.  ^ws.   xx'2  +  yy'2  =  1. 

19.    -,  +  ^  =  1.  20.    x«  +  r  =  l- 

\ii!}'        21.   What  are  the  equations  of  the  tangents  to  16,  17,  18,  20  at  the  point 
(1,  0)  ;  and  to  16,  18,  20  at  the  point  (0,  1)  ? 

Find  the  equation  of  the  tangent  to 
'   22.   y^  =  ix-S  x2,  at  the  point  (1,  1). 

23.  10  2/  =  (x  +  1)2  at  the  point  where  a;  =  9.     (Ex.  11,  p.  27.) 

24.  4  (x  +  l)  =  (y-  2)2  at  the  point  where  x  =  3.     (Ex.  11,  p.  27.) 

25.  (x  -  8)2  +  (2/  -  2)2  =  25  at  the  points  where  x  =  4. 

26.  x(x2  4-  2/2)  =  a(ic2  —  y^)  at  the  point  where  x  =  0,  and  ±  a. 

27.  Find  the  equation  of  the  tangent  to  [-]  4-(-)  =2,  and  show  that  at 
the  point  (a,  b)  it  is  the  same  for  all  values  of  n. 

Cf^'^-JS.   Show  that  the  curve  a;*  +  2/^  =  a^  becomes  steeper  as  it  approaches  the 
2/-axis,  and  is  tangent  to  the  axes  at  the  points  (±  a,  0)  and  (0,  ±  a). 

29.    Let  y=f{x)   and  y  =  F(x)   be  two  curves  intersecting  in  the  point 
(xi,  2/1) »  ^^^  l6t  0  be  the  angle  at  which  they  intersect.    Show  that 
tan0=   /^CxO-F^(xO, 

l+/'(Xi).i^'(Xi) 

What  is  the  condition  that  the  two  curves  shall  meet  at  right  angles?  be 
tangent  to  each  other  ? 

[The  angle  at  which  two  curves  intersect  is  the  angle  between  their  tangents 
at  the  point  of  intersection  of  the  curves.] 


80  SLOPE,   TANGENTS,   AND  NORMALS  [62 

30.  Find  the  angle  of  intersection  between  the  parabolas 

y'^  =  iax  and  x^  =  4  ay. 

31.  Show  that  the  confocal  parabolas 

y^  =  4  a{x  -h  a)  and  ^2  _  _  4  ^(^  —  b) 
intersect  at  right  angles. 

32.  At  what  angle  do  the  rectangular  hyperbolas 

y.^  _  yi  —  gp.  aj^(j  xy  =  b 

intersect  ?    Draw  several  sets  of  these  curves  by  assigning  different  values  to 
a  and  b. 

33.  Find  the  angle  at  which  the  circle  x^  +  y^  +  2x  =  12  intersects  the  parab- 
ola y^  =  9x. 

^       34.   Find  the  angle  of  intersection  between  x^  -\- y^  =  25  and  4  ?/2  =  9  x. 

>^       35.    Find  the  equations  of  the  tangent  and  the  normal  to  the  parabola  y^  =  'ix 
at  the  point  (4,  4). 

Also  find  the  angle  at  which  the  normal  meets  the  curve  at  its  other  point  of 
intersection  with  the  curve. 

36.  Find  the  derivative  of  the  quotient  of  two  functions. 

f(x) 
Xet  y  =  -'^  ^ ,  and  write  h  in  the  place  of  5x.  (1) 

(p(x) 

Then  ^  =  /"i    /r/(^±^_/Ml^a.  (2) 

dx     h  =  ^\l<p(x-]-h)      0(x)J         i  ^^ 

^   lim    r<f>(x)f(x  +  h)  -/(x)0(x  +  h)l  ,3^ 

^-OL  h(f>(x)(/>{x  +  h)  J 

Introducing  <f)(x)f(x)  —  <t>{x)f{x)  in  the  numerator  gives 

[x  +  h)-  <f>(x) 


d^       h  —  0  ]   


dx 


A •     (4) 


<p{x)(p(x  +  h) 


*L<|.(a?)J  [<|>(ic)]2 

Find  the  derivatives  of  the  following  functions : 
o-    ^  -  «  Ans.   _2A_.  38.  ^^  +  ^.  Ans.        "  ^ 


x  +  a  (x  +  a)2  x  +  l  (x  +  1)2 

39.   '-±^.  40.  -^ ^.     ^ns.       /^      ■ 

^j     a-2bx  ^2    __^!L_. 

(a  -  &x)2  ■    (1  -I-  x)« 


62]  SLOPE,   TANGENTS,   AND  NORMALS  81 

43.  Show  that  formula  (7),  §  60,  holds  (a)  when  n  is  a  negative  integer,  and 
(6)  when  «  is  a  rational  fraction. 

[To  prove  (a)  use  the  formula  in  Ex.  36  ;  for  (6)  use  (9),  §  61.] 

^    48.  Show  that  I>a.(taii  a?)  =  sec^  a?. 

Let  tf  =  tan  a;  =  ^^^.    Then  from  (5),  Ex.  36,  we  get 

cosx 

dy  _  cos  a;  •  Z>x(sin  x)  —  sin  x  •  Z)a;(cos  x) 
dx  cos2  JC 

But  Z>a;(sin  x)  =  cos  x,  and  i)x(cos  x)  =  —  sin  x.     [Ex.  5,  p.  75.] 
.  dy     cos2  X  +  sin2  x         1 


dx  cos2  X 

Prove  the  following  formulae : 


=  sec2  X. 


f-  49.  l>a.(cot  05)= -csc^o?.  ^  60.  I>a,(sec  »)  =  scc'a?  tan  a?. 

sj^    51.  l>a. (CSC  a?)  =  —  CSC  a;  cot  x,  ^  52.  Z>a.(sin  oc  cos  a?)  =  cos  2  x. 

Find  the  derivatives  of  the  following  functions  : 
/  63.  cos^  X.  54.  sin  X  —  4  sin^  x.  Ans.  cos^  x. 

65.  tanx-x.  se.  3tanx  +  tan3x. 

67.  X  sin  X  +  cos  x.  53^   ^^^3  ^  _  3  ^os  x.  Ans.  3  sin-3  x. 

60.    (rtx2  -}-  6)3.  ^ws.  6  ax{ax'^  +  &)2. 


59.  sec*  X  —  tan2  x. 
61.    (x2  +  a)(x2  +  6) 

63. 


^2_^2  62.    (a4-x3)(6  +  3x2). 

a^  +  a:2*  64.  tan2x  ^  ^  ^"^^^"^^•-  ^ns.  2sec22x. 

66.    («  +  x)V^^.  l-2sin2x 

^   .         g  66.  cot2x  =  Kcotx-tanx).     Ans.   -2csc-2x. 

a-  68.    (2x3+3)2(1-3x2)3. 


Vl4-x2  70.  sec x  + cosx.  

'0S2  X 
X3 

*^^'    (1  +  jca)2  *  72.  2  X  sin  x  +  (2  -  x2)  cos  x.  ^ws.  x2  s.    x. 

73      2  x2  —  1  sin"  .X     COS"*  X 

xv'l  +  x2  *   cos'"x      sin^x 


V:v,v^' 


y 


CHAPTER  VI 

THEORY  OF  EQUATIONS,    QUADRATURE,   AND  MAXIMA 
AND  MINIMA 

W\,\  THEORY  OF  EQUATIONS 

>J         63.   An  expression  of  the  form 

aa;"  +  &aj"-i  4-ca;'^-2  _| y-kx-^-l,  (1) 

where  n  is  a  finite  positive  integer  and  the  coefficients  a,  6,  c,  •  •  •  A:,  I 
do  not  contain  x^  is  called  a  Rational  and  Integral  Algebraic  Function 
of  X  of  the  nth  degree  ;  and 

ax""  +  6a;"-^+  ca;"-2  H \-'kx-\-l  =  ^  (2) 

is  called  the  General  Equation  of  the  nth  degree.  This  is  the  kind  of 
equation  we  shall  consider  in  this  section. 

If  we  divide  the  left  side  of  equation  (2)  by  a,  the  coefficient  of 
a;",  we  shall  obtain  the  genefal  equation  of  the  nth  degree  in  the 
standard  form, 

X-  -\-p,x^-'  -{-p^^-'  +  ...  +p^_^x  +p^  =  0,  (3) 

where  Pi,  P2,  "  •  Pn-i^  Pn  do  not  contain  x,  but  are  otherwise  unre- 
stricted. As  will  be  seen  hereafter,  some  of  the  properties  of 
equations  can  be  stated  more  concisely  when  the  equation  is  in  the 
standard  form. 

In  this  section  the  symbols /(ic), /i(a;),  (l){x),  <f)i(x),  etc.,  will  be  used 
to  denote  rational  integral  functions  of  x,  such  as  (1)  and  (3). 

Any  quantity  which  substituted  for  x  in  f(x)  makes  f{x)  vanish  is 
call    .  a  Root  of  f(x) ;  or  a  Root  of  the  Equation  f(x)  =  0. 

xf  we  put  y=f(x)  and  plot  the  locus  of  this  equation,  we  shall 
obtain  a  curve  which  is  called  the  Graph  of  f{x).  The  real  roots  of 
fix)  are,  therefore,  the  x  intercepts  of  its  graph.  ^~  ^ 

64.  A  rational  integral  function  ofx  is  continuous,  and  finite  for  ayiy 
finite  value  ofx. 


65]  THEORY  OF   EQUATIONS  83 

Let  f{x)  =  p^-+piaj"-i+p2»«-2+ ...  +p„_ia;+/)„.  (1) 

Then  each  term  will  be  finite,  provided  x  is  finite ;  and  therefore, 
as  the  number  of  terms  is  finite,  the  sum  of  them  all,  that  is/(ic), 
will  be  finite  for  any  finite  value  of  x. 

Now  suppose  X  receives  a  small  increment  ^,  producing  in  f{x)  the 
increment  f{x  -\-  li)  —f{x) ;  then 

+  "'+Pn-i[.{x-\-li)-x-\.  (2) 

Each  of  the  terms  in  the  right  member  of  (2)  will  become  indefi- 
nitely small  when  h  is  indefinitely  small ;  hence  their  sum  will 
become  indefinitely  small.  Therefore  f(x  -f  h)  —f(x)  can  be  made 
as  small  as  we  please  by  making  h  sufficiently  small.  This  shows  that 
as  X  changes  from  any  value  a  to  another  value  b,  f(x)  will  change 
gradually  and  without  interruption,  i.e.  without  any  sudden  jump, 
from  f(a)  to  f(b) ;  so  that  f(x)  must  pass  at  least  once  through  every 
value  intermediate  to  /(a)  and  /(&).  That  is,  f(x)  is  a  continuous 
function: 

Hence  the  graph  of  f(x)  is  a  continuous  curve  with  finite  ordinates 
for  finite  values  of  x. 

65.    To  calculate  the  numerical  valv^  off(a). 

Let  f{x)  =p^  -\-p^x^  ^-p>pi  4-^3-  (1) 

Then  we  wish  to  calculate  the  numerical  value  of 

/(a)  =po«'  +p^a^  -\-p2a  +ps.  (2) 

This  result  is  most  easily  obtained  as  follows : 

Multiply  Pq  by  a  and  add  to  pi,  this  gives  poa  -{-pi ; 

Multiply  this  by  a  and  add  to  j72,  this  gives  Pffi^  -\- p^a  -\-  P2', 

Multiply  this  by  a  and  add  topg,  this  gives  Poa^  +i>ia^  +i?2«  +pt' 

The  process  may  be  arranged  in  the  following  way : 

Po  Pi  P2  Ps 

Po« Poa^-\-Pia poa^+pia^+p^ 

Po  i>oa+i>i  Poa^-hl)ia-\-p2  Poa^  +  Pia^  +  p^a -\- ps. 


84  THEORY   OF   EQUATIONS  [66 

We  may  proceed  in  the  same  way,  whatever  the  degree  of /(x). 

Ex.    Find  the  numerical  value  of  /(3)  if 

/(a;)  =2  X*- 7x3  +  13  a^  _  16. 


2 

-7 

0 

13 

-16 

6 

-3 

-9 

12 

V 

-1 

-3 

4 

-4 

••  /('^) 

=  -4. 

is 

process 

is 

called 

Synthetic  1 

Substitution. 

66.    To  find  the  remainder  and  the  quotient  when  f(x)  is  divided  by 
X  —  a,  where  a  is  any  constant. 

Divide /(a?)  by  a;  —  a  until  the  remainder  no  longer  contains  x. 
Let  fj>{x)  denote  the  quotient  and  R  the  remainder.     We  then  have 
the  identical  equation 

f{x)  =  <ly{x){x-a)-\-R,  (1) 

which  must  be  satisfied  when  any  value  whatever  is  substituted  for 
X.     Let  ic  =  a,  then 

f{a)  =  ^(a){a-a)+It  =  It;  (2) 

for  (ji  (a)  (a  —  a)=  0,  since  by  §  64  <^  (a)  is  finite.  That  is,  the 
remainder  is  equal  to  the  result  obtained  by  substituting  a  for  x 
in  the  given  function. 

CoR.    If  3.  is  a  root  off(x),  thenf(x)  is  divisible  by  x  —  sl. 

Conversely^  iff(x)  is  divisible  by  x  —  sl,  then  a  is  a  root  off{x). 

For,  if  either  f(a)  =  0,  or  E  =  0,  in  (2)  the  other  is  also  equal  to 
zero,  which  proves  the  proposition. 

Let  f(x)  =  PqX^  -{-p^x^  +P2^  +  jPs,  for  example. 

By  actual  division  we  find 

<f>(x)  =P(fl^-h  (poa  -\-p;)x  4-  (Poa^  +i>ia  +P2), 
and  R  =  p^a^  -f-  p^a"^  4-  p^ci  +  p^. 

By  comparing  these  expressions  with  the  results  found  in  §  65 


67]  THEORY  OF   EQUATIONS  86 

we  see  that  R  and  the  coefficients  in  <^(cc)  are  the  same  as  the  sums 
obtained  by  synthetic  substitution. 

Ex.     Find  0(x)  and  B  when  3  x^  —  2  x*  —  16  x^  -  x  +  7  is  divided  by  (x  +  2). 
3         _2         -10        0         -1         +7  -^  _ 

-6+100  0+2 


3         -8  U        0         -I         +9 

Thus  0(x)  =  3  X*  -  8  x3  -  1,  and  i?  =  9. 

.  •.     3  x5  -  2  x*  -  16  x3  -  X  +  7  =  (x  +  2)  (3  x*  -  8  x^  -  1)  +  9. 

This  process  can  be  applied  to  any  function  of  any  degree,  and 
is  a  particular  case  of  Synthetic  Division.  (See  Todhunter's 
Algebra,  Chap.  LVIII.) 

67.   An  equation  of  the  nth  degree  has  n  roots,  real  or  imaginary. 
Let  the  equation  be 

fix)  =  a;"  +piaj"-^  +^2^""^  +  •••  +  Pn  =  0.  (1) 

Let  «!  be  one  root*  of  the  equation /(ic)  =  0,  then/(.c)  is  divisible 

by(a;-aO.     (^66.) 

.'.    f{x)  =  {x-a,)f{x),  (2) 

where  f{x)  is  an  integral  function  of  x  of  degree  (w  —  1). 
In  like  manner  if  ag  is  a  root  of  /i(a;),  then 

f{x)  =  {x-a,)f^{x\  (3) 

where /2(a;)  is  an  integral  function  of  x  of  degree  {n  —  2). 

Proceeding  in  this  way  we  shall  find  n  factors  of  the  form 
{x  —  a^),  and  we  have  finally, 

f(x)  =  (qc  —  ay)  ipc  -  at)  (a?  -  as)  •••  (»  -  a«)  =  0.  (4) 

It  is  now  clear  that  ai,  as,  otg  .  .  .  a„  are  roots  of  the  equation 
f{x)  =0;  and  as  no  other  value  of  x  will  make  fix)  vanish,  the 
.equation  can  have  no  other  roots. 

The  factors  of /(ic)  need  not  all  be  different  from  one  another; 
thus  we  may  have 

*  We  here  assume  the  fundamental  theorem  that  every  equation  has  one  root,  real  or 
imaginary.  Proofs  of  this  theorem  have  been  given  by  Argand,  Cauchy,  Clifford,  and 
others,  but  they  are  too  difficult  to  be  included  in  this  book.  The  student,  however,  is 
already  familiar  with  the  fact  that  every  equation  of  the  first  degree  has  one  root; 
that  every  equation  of  the  second  degree  has  two  roots,  real  or  imaginary ;  and  it  will 
be  shown  in  §  71  that  every  equation  of  an  odd  degree  has  oive  real  root. 


86  THEORY  OF   EQUATIOISTS  [67 

f(Qc)  =  (Qc-a\)p(pc-a2Yidc-azy-",       "  (5) 

where  p  -\-q  +  r  -\-  •  •  •  =  7i. 

In  this  case  f{x)  has  p  roots  each  a^,  q  roots  each  ag,  etc.,  the  whole 
number  of  roots  being 

p  +  q  +  r  -\-  •••  =n. 

Therefore  the  graph  of  f{x)  will  cut  the  x'-axis  in  n  points,  which 
may  be  real,  coincident,  or  imagmary:    and  the  real  roots  are  its 
i^ntercepts. 
f  (       Hence  the  real  roots  of  a  function  may  be  found  exactly  or  approxi- 
mately by  constructing  its  graph. 

EXAMPLES 

1.  Divide  2  a;^  -  6  a:*  -  5  ic2  +  10  x  +  18  by  x  -  3. 
Find  the  other  roots  of  the  following  equations  : 

'  2.  Two  roots  of  x*  -  12  x^  +  49  x2  -  78  x  +  40  rr  0  are  1  and  5. 

3.  One  root  of  x^  -  16  x2  +  20  x  +  112  =  0  is  -  2. 

4.  Two  roots  of  x*  +  8  x^  -  22  x2  -  16  x  +  40  =  0  are  2  and  -  10. 

5.  Two  roots  of  x*  -  12  x^  +  48  x2  -  68  x  +  15  =  0  are  5  and  3. 

6.  Three  roots  of  6  x^  +  11  x*  -  21  x^  +  7  x^  +  15  x  -  18  =^  0  are  ±  1  and  -  3. 

Find  graphically  the  exact  or  approximate  roots  of 

7.  x3-2x2-llx+  12  =  0. 

8.  x4  -  8  x3  +  14  x2  +  8  X  -  15  =  0. 

9.  x*  -  2  x3  -  13  x2  -  14  X  +  24  r=  0. 

10.  x3  -  8  x2  -  28  X  +  80  =  0. 

11 .  6  x3  -  13  x2  -  21  X  +  18  =  0. 

12.  8x^-18  x2- 71  x  + 60  =  0. 

13.  x*-6x3-5x2  +  56x-30  =  0. 
Form  the  equations  whose  roots  are 

14.  1,3,-5.  15.  -2,  3,  -4,  6. 

16.  h  -  I,  f.  17.  ±  1,  ±  4. 

18.  0,  1,  -4,  5.  19.  ±  V2,  ±  V3. 

20.  0,  -  2,  ±  V^^.  21.  3,  5  ±  V^- 

22.  4  ±  V3,  -  1  ±  v/6-  23.  1,  -  2,  3,  -  4,  5. 

24.  0,  2  ±  V^^,  -  3  ±  y/Q.  25.  0,  0,  \,  -  f,  1  ±  V2. 

26.  1  ±  V^=^,  -  2  ±  V^.  27.  -  3,  2  ±  V-Z,  -  3  ±  V^^. 


68]  THEORY   OF   EQUATIONS  87 

68.   Relations  betiveen  the  roots  and  the  coefficients  of  an  equation. 
If  there  are  two  roots,  a^  and  ag,  we  have  (§  67) 
x^-hPiX-\-p2  =(x-  ai)(x  —  a^ 

=  x^—  (ai  +  a^^x  +  aya^.  ~~  (1) 

.-.  «!  4-  ttg  =  — pi,     a^a^  =p2. 
If  there  are  three  roots  a^,  a^^  and  a^,  we  have 
yf'  +  pior  H-  p^  +p^={x—  ay){x  —  a^{x  —  a^) 

=  a^  —  (tti  +  tta  4-  ag)^.-^  4-  (aiCtg  +  «2  «3  +  cisai)x  —  a^a^a.^.     (2) 
.-.     Oi  4-  ttg  +  ttg  =  —  j9i,     aia2  +  «2«3  4-  ^s*^!  =  i^2j     «i«2a3  =  —  i>3- 
In  like  manner  if  the  equation  is  of  the  nth  degree  and  therefore 
has  n  roots  a^,  ag  •••  a,.  •••  a„,  then 
a;"  4-i)i^"-'  +i>2^"-'  4-  •••  H-p.a;''-'-  4-  •••  +i)„ 

=  {x  —  ay){x  —  as)  •••  {x  —  a,)  •••  («  —  a„)  (3) 

=  a;**  -  S,x^-^  4-  /Saa)"-^ +  (  -  1)'->S'X"*" 

±-+(-ir^n,  (4) 

where  S,.  is  the  sum  of  all  the  products  of  aj,  a2,  •"  a^"-  a^  taken 
r  together. 

Equating  the  coefficients  of  the  same  powers  of  x  on  the  two  sides 
of  the  identity  (4)  gives 

Sl  =  -Pu    S2-=P2,    Sr=(-lVPr9 
Sn  =  i-  ^)^Pn  =  «l«2  •••  «r  •.•  «n. 

//*  Pn  =  0,  one  rooi  is  zero  ;  if  p^  =  p„_i  =  0,  tivo  roots  are  zero  ;  if 
Pn  =  Pn-i  =  •  •  •  Pn-r  =  0,  r  +  1  roots  are  zero. 

EXAMPLES 
Find  the  other  roots  of  the  following  equations : 

1.  Two  roots  of  a;3  +  a;2  —  4  X  —  4  =0  are  2  and  —  1. 

2.  Two  roots  of  a;^  _  4  a;2  _  3  a;  +  12  =  0  are  4  and  ^3. 

3.  Two  roots  of  ic^  -  13  a;  +  12  =  0  are  1  and  3. 

4.  Three  roots  of  x*  -  10  x*  +  35  a;^  -  50  x  4  24  =  0  are  1 ,  2,  and  3. 

5.  One  root  of  x»  -  6  a:^  ^  12  x^  =  0  is  3  -  V  -  3. 

6.  Two  roots  of  6  ar*  -  7  a:^  _  14  ^2  .{-  15  ^  =  0  are  1  and  f . 

7.  Two  roots  of  4  xf^  -  5  x*  +  2  x^  4  6  x2  =  0  are  1  ±  V^HT 


88  THEORY  OF   EQUATIONS  [69 

69.    Tlie  first  term  of  f  (x)  can  he  made  to  exceed  the  sum  of  all  the 
other  terms  by  giving  to  x  a  value  sufficiently  great. 

Let  fix)  =  ii^x"  +pi.c'*-^  +^2^*'"^  H VPn, 

and  let  k  be  the  greatest  of  the  coefficients ;  then 


\i^ 


_  p^x'Xx  -  1)      p^\x  -  1)  ^  Po .        .  . 
~   k{x''-l)   -^        kx''  k^         ^' 

Now  %^(x  —  l)  can  be  made  as  great  as  we  please  by  sufficiently 
k 

increasing  x,  which  gives  the  proposition. 

70.  An  even  number,  or  an  odd  number,  of  real  roots  of  f(x)  =  0 
lie  between  a  and  b  according  as  f(a)  and  f(b)  have  the  same  sign,  or 
opposite  signs. 

The  two  points  A[a,  /(a)]  and  B[b,  /(6)]  are  on  the  same  side,  or 
on  opposite  sides,  of  the  a>axis  according  as  /(a)  and  f(b)  have  the 
same  sign,  or  opposite  signs. 

Therefore,  since  the  graph  of  f{x)  is  a  continuous  curve  (§  64),  in 
passing  from  ^  to  B  along  the  graph  the  ic-axis  will  be  crossed  an 
even  number,  or  an  odd  number,  of  times  according  as  f(a)  and  f(b) 
have  the  same  sign,  or  opposite  signs.  This  proves  the  proposition. 
(An  even  number  includes  the  case  of  ?io  roots.) 

■E.g.   If /(x)  =x^-Zx  +  \,  then /(I)  =  -  1  and/(2)  =  3. 

.-.    At  least  one  real  root  ofa;^  —  3x  +  l=:0  hes  between  1  and  2. 

71.  An  equation  of  an  odd  degree  has  at  least  one  real  root. 
Let  the  given  equation  be 

f{x)  =  x^--^^  +Pia^"  4-JP2aj'"-'  +  •  •  •  +i^2n+i  =  0. 

Let  a  be  a  positive  value  of  x  sufficiently  large  to  make  the  first 
term  of  f{a)  greater  than  the  sum  of  all  the  other  terms  (§  69). 
Then  the  sign  of  f(a)  will  be  the  same  as  the  sign  of  a-"+^,  i.e.  the 
same  as  the  sign  of  a. 

Hence  if  a  be  sufficiently  great,  /(a)  is  positive,  /(O)  =_P2n+ij  and 
f(  —  a)  is  negative. 

Therefore  in  all  cases  there  is  one  real  root,  which  is  positive  or 
negative  according  as  p2n+i  is  negative  oi  positive  (§  70). 


78] 


THEORY  OF   EQUATIONS 


89 


Hence  the  graph  of  a  fiinctioii  of  an  odd  degree  in  the  standard  form 
extends  to  infinity  in  the  first  and  third  quadrants. 

72.  An  equation  of  an  even  degree  in  the  standard  form  with  the 
last  term  negative  has  at  least  tivo  real  roots  with  opposite  signs. 

Let  the  given  equation  be  ~ 

f(x)  =  x'-  +i>iar'"-'  +i>2a^-'"-'  +  •••  +  P'm  =  0. 

If  a  is  taken  sufficiently  great,  f(a)  will  have  the  same  sign  as 
a'^"(§  69),  which  is  positive  for  both  positive  and  negative  values 
of  a;  that  is, /(a)  and/(—  a)  will  both  be  positive,  while /(O)  =  2>2n; 
which  by  hypothesis  is  negative.  Therefore  there  is  at  least  one  real 
root  between  0  and  a,  and  another  between  0  and  —  a  (§  70). 

Tlie  graph  of  a  function  of  an  even  degree  in  the  standard  form  ex- 
tends to  infinity  in  the  first  and  second  quadrants. 

73.  To  find  approximately  the  real  roots  of  f{x)  =  (). 

Plot  the  graph  of  f{x)  and  thus  find  the  pairs  of  numbers,  usually 
consecutive  integers,  between  each  of  which  one  root  lies. 


Suppose  /(a)  =  CA,  a  positive  number ;  and  f{a  +  1)  =  DB,  a 
negative  number. 

Then  there  is  at  least  one  real  root  (§  70)  between  a  and  a  -h  1. 

Draw  the  chord  AB  cutting  the  a;-axis  in  E ;  draw  BF  parallel  to 
the  a?-axis  meeting  AC  produced  in  F. 


90  TIIP:0RY   of   equations  [74 

Then,  if  there  is  only  07ie  root  between  a  and  a  +  1,  it  is  approxi- 
mately equal  to  OE ;  if  the  graph  were  a  straight  line,  it  would  be 
exactly  equal  to  OE. 

Since  the  triangles  ACE  and  AFB  are  similar,  and  FB  =  1, 

^^^FB^CA^       CA       ^  f(a)  .^. 

FA  CA  +  BD     f(a)  -  f(a  +  1) 

If  we  use  numerical  values  of  f{d)  and  f(a  -f  1),  we  shall  then 
have  for  all  cases 

OE  =  a  + f~^^ *  (2) 

Ex.     Find  the  roots  of  x^  -  29  a;  +  42  =  0. 

Here  /  (4)  =  —  10  and  /  (5)  =  22.     Hence  there  is  a  root  between  4  and  5. 

Substituting  in  (2)  gives         OE  =  4  +  — — —  =  4.4  -. 

"       ^  ^  ^  10  +  22 

Then  /(4.4)  =  -  .416  and  /(4.5)  =  2.625. 
Hence  the  root  lies  between  4.4  and  4.5. 

When  the  root  is  greater  than  OE,  as  in  the  diagram  and  also  in  this  example, 
it  is  better  to  try  the  figure  next  greater  than  that  given  by  the  quotient. 
The  next  figure  of  the  root  may  now  be  approximated  in  the  same  way. 

Thus  f(iA)x.l       ^-OilG^  Q^    gj  ^^-^  ^ 

/(4.4)+/(4.5)      3.041 

.*.   The  approximate  root  is  4.41.     The  exact  root  is  (3  +  ^^2). 

EXAMPLES 

Calculate  to  two  places  of  decimals  the  real  roots  of  the  equations 

1.  a;3  -  3  a:  -  1  =  0.  6.  x*  -  12  x  +  7  =  0. 

2.  ccS  -  7  ic  +  7  =  0.                                     7.  a:*  -  5  a;3  +  2  x2  -  13  a;  +  55  =  0. 
Z.    x^  +  2x'^  -Sx-9  =  0.                           8.  a;3-3x2-2a;  +  5  =  0. 

4.  x3  +  2  x2  -  4  a;  -  43  =  0.  9.  a;^  -  81  a;  +  40  zir  0. 

5.  a:3  -  15  X  +  21  =  0.  10.  x^  -  55  x2  -  30  x  +  400  =  0. 

74.  In  any  equation  with  real  coefficients  imaginary  roots  occur  in 
pairs. 

I.  Let  f(x)  =  0  be  an  equation  with  real  coefficients  having  r  real 
roots  and  the  other  roots  imaginary.     Then 

f{x)  =  (x-  a,)  (x  -  a,)  ...  (x  -  a,)<l>(x)  =  0,     (§  67)  (1) 

*  The  student  should  compare  this  method  with  Horner's  Method  of  Approximation 
found  in  almost  any  complete  algebra. 


74]  THEORY  OF   EQUATIONS  91 

where  <f>(x)  is  a  function  with  real  coefficients  whose  roots  are  all 
the  imaginary  roots  of  f{x),  and  no  others.  Hence  <f)(x)  must  be 
of  even  degree,  and  therefore  has  an  even  number  of  roots.  Other- 
wise it  would  have  at  least  one  real  root  (§  71). 

Therefore  (1)  has  an  even  number  of  imaginary  roots.  _ 

II.  If  a  +  6V—  1  is  a  root  of  an  equation  with  real  coefficients, 
then  a  —  bV  —  lis  also  a  root. 

Let  the  equation  be 

a;"  +pix^-''  +i92a5"-2  +  ...  +p„  =  0.  (2) 

Substituting  a  -f-  6V—  1  for  a;  in  (2),  we  have 

(a  -h  bV^^y+2h{a  +  bV^^y-^  +P2(a  +  bV^ly-^ 

-\-"'+Pn  =  0.  (3) 

Expanding  by  the  binomial  theorem,  and  collecting  together  the 
real  and  imaginary  terms,  we  shall  have  a  result  in  the  form 

P+QV^1  =  0.  (4) 

In  order  that  this  equation  may  hold  we  must  have 

P=Q  =  0.  (6) 

Since  P  and  Q  are  real,  they  contain  only  even  powers  of  V—  1, 
and  hence  will  not  be  changed  by  changing  the  sign  of  V—  1. 
Therefore,  when  a  —  foV—  1  is  substituted  for  x  in  (2),  the  result 
willbeP-QV^^.  

But  from  (5)  p _  Q V-  1  =  0. 

.-.  a  —  6V—  1  is  also  a  root  of  (2). 

Corresponding  to  the  roots  a  ±  6  V—  1  of  f(x)  =  0,  f(x)  will  have 
the  real  quadratic  factor  l(x  —  ay  +  b^]. 

The  two  quantities  a  ±  ftV—  1  are  called  conjugate  imaginary  ex- 
pressions. 

Show  that  the  locus  of  the  equation  y  =  x^-^k  cuts  the  a^axis 
in  two  points  which  are  real  and  distinct,  real  and  coincident,  or 
imaginary  according  as  k  is  negative,  zero,  or  positive.  Hence 
illustrate  graphically  the  preceding  theorem  by  showing  that,  as 
the  absolute  term  of  f(x)  is  changed,  real  intersections  of  its  graph 


92  THEORY  OF  EQUATIONS  [75 

with  the  ic-axis  disappear  or  reappear  in  pairs;  and  that  the  pas- 
sage from  a  pair  of  real  distinct  roots  to  a  pair  of  imaginary  roots 
is  through  a  pair  of  real  coincident  roots. 

EXAMPLES 

1.  Show  that  if  either  a  ±^b  is  a  root  of  an  equation  with  rational  co- 
efficients, the  other  is  also  a  root. 

2.  Solve  the  equation  x^  —  2x^  —  22x^  +  &2x  —  15  =  0,  having  given  that 
one  root  is  2  -f  ^^3. 

3.  Solve  the  equation  2x3  -  ISac^  +  46x  -  42  =  0,  having  given  that  one 
root  is  3  +  V—  5. 

4.  If  y/a  +  y/b  is  a  root  of  an  equation  with  rational  coefficients,  y'a  and 
■y/b  not  being  similar  surds,  show  that  ±■^ya  ±^b  will  all  four  be  roots. 

5.  Form  the  biquadratic  equation  with  rational  coeflBcients  one  root  of  which 
is  V2  +  V^- 

6.  Show  that  Ex.  4  holds  when  either  or  both  a  and  b  are  negative. 

7.  Find  the  biquadratic  equation  with  rational  coefficients  one  root  of  which 
is  V2  +  V^=~3. 

8.  Solve  the  equation  2x^  -  Sxp  -\-  5x^  +  Qx^  -  27x  +  81  =  0,  having  given 
that  one  root  is  ^2  +  V—  1. 

Transformation  of  Equations 

75.  To  find  an  equation  whose  roots  are  those  of  a  given  equation 
with  opposite  signs. 

If  the  given  equation  is  f{x)  =  0,  the  required  equation  will  be 
/(  —  a;)  =  0.  For,  when  x  =  a,f(x)  =f{a),  and  when  x  =  —  a,f{  —  x) 
=f{a) ;  hence,  if  a  is  a  root  of  f{x)  —  0,  then  —  a  will  be  a  root  of 
/(-x)  =  0. 

The  graph  of  /(  —  x)  is  the  reflection  of  the  graph  of  f{x)  in  a 
mirror  through  the  ?/-axis  perpendicular  to  the  plane;  i.e.  the  two 
graphs  are  symmetrical  with  respect  to  the  ?/-axis,  which  proves  the 
transformation  for  real  roots. 

li  f{x)  =f(-x)  [§  28,  (2)],  the  two  graphs  will  coincide,  and 
the  roots  otf(x)  will  occur  in  symmetric  pairs  of  the  form  ±  a. 

The  transformed  equation  is  found  by  simply  changing  the  signs 
of  all  the  terms  of  odd  degree,  or  of  all  the  terms  of  even  degree,  in 
the  given  equation. 


76]  THEORY  OF  EQUATIONS  93 

76.  To  find  an  equation  whose  roots  are  those  of  a  given  equation, 
each  diminished  by  the  same  given  quantity. 

If  we  put  x  =  x'  -{•  hj  the  origin  will  be  moved  to  the  right  a 
distance  equal  to  h  [§  53,  (1)]. 

Hence  the  avintercepts  of  the  graph  of  f(x),  i.e.  the  real  roots 
of  f(x),  will  each  be  diminished  by  h. 

Therefore,  if  f(x)  =  0  is  the  given  equation,  the  required  equation 
will  be  f(x-{-h)=0.  For,  when  x  =  a,  f(x)=f(a),  and  when 
X  =  a  —  h,f(x -{- h)  =f(a)',  hence,  if  a  is  a  root  of  /(a?)  =  0, 
then  a  —  ^  is  also  a  root  of  f(x  -f-  h)  =  0,  whether  a  is  real  or 
imaginary."*^ 

The  coefficients  of  the  new  equation  can  be  found  by  synthetic 
substitution  as  follows : 

Ex.  Transform  the  equation  cc*  —  3  a;^  —  15  x^  +  49  a;  —  12  =  0  into  another 
whose  roots  shall  be  those  of  the  first  each  diminished  by  2. 

1         -  3         -  15         +49         -  12 
^Pei'ation  2         -2         -34         +30 


1 

-1 

2 

-17 
2 

+  15 
-30 

+  18 

1 

+  1 
2 

-15 
+    6 

-16 

1 

+  3 

2 

-   9 

1  5 

.'.   a;*  +  5  x^  —  9  x2  —  15  a;  +  18  =  0  is  the  required  equation. 
[Check  this  result  by  substituting  directly  a;  +  2  for  x.] 

If  we  put  x  =  x'  —  — ,  where  »i  is  the  coefficient  of  a;**~^,  each  root 
n 

will  be  diminished  by  (  — -^  i,  and  therefore  the  sum  of  the  roots 


will  be  diminished  by  n  f  —  ^]=  —p^, 

\      nj 


Hence  the  sum  of  the  roots  of  the  new  equation  will  he  zero  (§  68)'; 
le.  the  coefficient  of  the  second  term  ivill  be  zero, 

Ex.  lYansform  the  equation  x'J  +  0x2  +  4x  +  5  =  0  into  another  in  which  the 
coefficient  of  x^  is  zero. 

*  This  transformation  is  used  iu  Horner's  Method.    See  foot-note,  p.  90. 


94  THEORY  OF  EQUATIONS  [77 

Let  x  =  x^  —  2,  since  pi  =  6  and  w  =  3 ;  then  we  obtain 
1         +6         +4         +5 


-2 

-8 

+    8 

1 

+  4 
-2 

-4 
-4 

+  13 

1 

+  2 
-2 

-8 

1  0 

.  •.   ic^  —  8  jc  +  13  =  0  is  the  required  equation. 

77.  To  find  an  equation  whose  roots  are  the  reciprocals  of  the  roots 
of  a  given  equation. 

Let  the  given  equation  be 

P^^  +  p^X^-^  +  P2^"-'  +    •  •  •    +  Pn-V«  +Pn=0-  (1) 

Substituting  -  for  x  in  (1)  gives 

z 

i)»+,,Q"-V^,Q"-V  ...  +„„_.(r)+^„  =  0,      (2) 

which  is  the  required  equation,  for  (2)  is  satisfied  by  the  reciprocal 
of  any  quantity  which  satisfies  (1). 
Multiplying  (2)  by  z'^  gives 

K2"  +  Pn-i^""-'  +  Pn-^""-'  +  '"  -{-PiZ+Po==0.  (3) 

Therefore  the  required  equation  is  obtained  by  merely  reversing  the 
order  of  the  coefficie7its  of  the  given  equation. 

If  p^^  =  0,  one  root  of  (1)  is  zero,  and  hence  the  corresponding  root 
of  (2)  is  infinite.  Therefore,  as  the  coefficient  of  the  highest  poiver  of 
X  in  f{x)  ap])roaches  the  limit  zero,  07ie  root  of  fix)  becomes  infinite. 

If  the  coefficients  of  (1)  are  the  same  (or  differ  only  in  sign)  when 
read  in  order  backwards  as  when  read  in  order  forwards,  the  roots  of 
(1)  and  (3)  are  the  same.     That  is,  the  roots  of  (1)  will  then  occur 

in  pairs  of  the  form  a  and  -• 

a 
An  equation  in  which  the  reciprocal  of  any  root  is  also  a  root  is 
called  a  Reciprocal  Equation. 

E.g.  C  a;3  _  19  ^2  _|_  19  -y  _  6  _  0  is  a  reciprocal  equation  in  which  the  coeffi- 
cients differ  in  sign  when  read  in  order  backwards  and  forwards  ;  two  roots 
are  f  and  f . 


77]  THEORY   OF   EQUATIONS  95 

EXAMPLES 

Find  the  equations  whose  roots  are  those  of  the  following  equations  with 
op{)osite  signs  : 

1.   ic2  -  4  a;  -  5  =  0.  2.  ,x^  +  6x^  -  7  x  -  GO  =  0. 

3.   x8  -  8  X--2  -  28  X  +  80  =  0.  4.  a:*  -  12  x"^  +  12  x  -  3  =  0.     -  - 

Find  the  equation  whose  roots  are  those  of 

6.   x3  -  16  x2  +  20  X  +  112  =  0,  each  diminished  by  4. 

6.  X*  -  12  x3  +  49  x2  -  78  X  +40  =  0,  each  diminished  by  2. 

7.  x*  -  3  x8  -  6  x2  +  14  X  +  12  =  0,  each  diminished  by  -  2. 

Transform  the  following  equations  so  as  to  make  the  second  terms  dis- 
appear : 

8.  x2  -  4  X  -  21  =  0.  9.   x3  -  6  x2  +  8  X  -  2  =  0. 

10.  X*  +  4  x8  -  29  x2  -  156  x  +  180  =  0. 

11.  Find  the  equation  whose  roots  are  those  of  x^  +  6  x2  —  15  x  +  12  =  0 
each  diminished  by  c,  and  find  what  c  must  be  in  order  that,  in  the  trans- 
formed equation,  (1)  the  sum  of  the  roots,  and  (2)  the  sum  of  the  products 
of  the  roots  two  together,  may  be  zero. 

12.  Transform  the  equation  x^  +  3  x2  —  9  x  —  27  =  0  into  another  in  which 
the  coefficient  of  x  shall  be  zero. 

Find  the  equation  whose  roots  shall  be  the  reciprocals  of  the  roots  of 

13.  x2  -  8x  -  9  =  0.  14.    2  x3  +  3  x2  -  13  X  -  12  =  0. 
16.    6  X*  -  5  x3  -  30  x2  +  20  X  +  24  =  0. 

16.  Show  that  a  reciprocal  equation  of  an  odd  degree  whose  corresponding 
coefficients  have  the  same  sign  has  one  root  equal  to  —  1. 

17.  Show  that  a  reciprocal  equation  of  an  odd  degree  in  which  corresponding 
coefficients  have  opposite  signs  has  one  root  equal  to  4- 1. 

18.  Show  that  a  reciprocal  equation  of  an  even  degree  in  which  correspond- 
ing coefficients  iiave  opposite  signs  has  the  two  roots  ±  1. 

Solve  the  following  equations : 

19.  2  x3  -  7  x2  +  7  X  -  2  =  0.  20.    0  x^  -  7  x2  -  7  x  +  6  =  0. 
21.    3  x^  +  5  x2  -I-  5  X  +  3  =  0.  22.    5  x=^-  7  x'^  +  7  x  -  5  =  0. 

23.   2  X*  +  5  x=^  -  5  X  -  2  =  0.  24.    12  x*  -  25  x»  +  25  x  -  12  =  0. 

25.  6  x*  -  7  x«  +  7  X  -  6  =  0. 

26.  Solve  the  equation  2  x*  -  3  x^  -  16  x2  -  3  x  +  2  =  0,  having  given  that 
one  root  is  —  2. 

27.  Solve  the  equation  14  x^  -  3  x<  -  34  x^  -  34  x2  -  3  x  +  14  =  0,  having 
given   that  one   root  is  2. 

28.  Solve  the  equation  10  x'^  -  21  x-'  +  21  x  -  10  =  0,  having  given  that  one 
root  is  2. 


96 


THEORY  OF   EQUATIONS 


[78 


78.  Successive  Derivatives.  If  f(x)  denote  any  function  of  x, 
its  derivative  f'(x),  (§  58),  will  in  general  be  a  function  of  x  that 
can  also  be  differentiated.  The  result  of  differentiating  f'{x)  is 
called  the  Second  Derivative  of  f{x).  If  this,  again,  can  be  differ- 
entiated, the  result  is  called  the  Third  Derivative,  and  so  on. 

The  successive  derivatives  of  f(x)  will  be  denoted  by 

n^),  /"(^),  f"'(^)  -/"H^). 

Let  f(x)  =Aq  +  Aix  -f-  A.2x'  +  A^x^  -{-...+  A,,x''. 

Then       /' (x)  =  A,  i- 2  A^ -\- 3  A^x"  +  •  •  •  +  wyIX"',  ( §  60) 

/"(ic)  =  2  ^2  +  2  .  3  ^3^;  +  •••  +  w(n  -  1)  A^x^-\ 
f"(x)  =  1.2. 3  ^3+  •••  +  n{n-l)  (n  -  2)  A^x^'-'', 


f^^\x)  =  n{n  -l){n  -  2)  ...  3  •  2  •  1  /!„  =  A  *  ** !  • 
E.g.  if  f(x)  =x4  -  3  a;3  -  5  x'-^  +  2  X  -  1, 
then         /'(x)  =4a;3-9a:2- lOx  +  2,  /'"(a:)  =  24  a:  -  18, 

ff'{x)  =  12  a;2  -  18  a;  -  10,  f""(x)  =  24  =  4  ! . 

Hence  the  ^-th  derivative  of  a  rational  integral  function  of  the 
nth  degree  is  itself  a  rational  integral  function  of  degree  {n  —  r), 
(where  r  is  not  greater  than  n)  ;  and  the  nth.  derivative  is  a  con- 
stant. Therefore  the  preceding  theorems  pertaining  to  a  rational 
integral  function  f{x)  will  also  hold  for  its  derivatives, 

79.    The  Derivative  Curve,  and  Elboivs. 

Y 


Let   the   curves  LM  and  L'M'  be  the  loci,  respectively,  of  the 
equations 

y=f(^)  (1) 

and                                              y=f\x).  (2) 


79]  THEORY  OF  EQUATIONS  97 

We  will  call  VM\  the  locus  of  (2),  the  Derivative  Curve,  (or 
D.  C),  and  LM  the  Integral  Curve.     (See  §  81.) 

Draw  any  line  parallel  to  the  7/-axis  meeting  the  a>axis  in  Q,  and 
the  curves  in  P  and  P\ 

We  will  call  P  and  P'  corresponding  points.  ~  ~ 

Then,  if  OQ  =  a,  we  have  by  §  58 

qp  =f(a)  =  slope  ofLMat  P. 

Hence  the  D.  C.  is  a  curve  such  that  its  ordinate  at  any  point  is 
the  slope  of  the  integral  curve  at  the  corresponding  point. 

Let  A,  B,  C,  D  be  the  points  on  LM  where  the  slope,  i.e.  f\x),  is 
zero ;  then  the  ordinates  of  the  corresponding  points  A\  B',  C,  D' 
on  L'M'  are  zero.  Hence  A',  B',  C,  D'  are  the  intersections  of  V3r 
with  the  cc-Sixis.  Between  A  and  B  the  slope  of  LM  is  positive, 
between  B  and  C  negative,  etc.  Therefore,  between  A'  and  B'  the 
curve  L'M'  is  above  the  a>axis  between  B'  and  C  below,  etc. 

It  will  be  convenient  to  call  such  points  as  A,  B,  C,  D,  Elbows 
of  the  curve.  Then  the  abscissas  of  the  elbows  of  the  graph  of 
f(x)  are  the  roots  of  /'(«),  and  may  therefore  be  found  by  plotting 
the  D.  C.  or  by  solving  the  equation  f'(x)  =  0. 

Since  /  (x)  is  of  degree  {n  —  1),  (§  78)  the  graph  of  f(x)  cannot 
have  more  than  (n  —  1)  elbows. 

If  f(x)  is  of  an  odd  degree,  its  graph  will  have  an  even  number 
of  elbows  (including  no  elbows),  and  therefore  f(x)  will  have  at  least 
one  real  root.     (Cy.  §  71.) 

If  the  roots  of  /'  (x)  are  all  imaginary,  the  graph  of  f(x)  will  have 
no  elbows. 

If  two  roots  of  f'(x)  are  equal,  its  graph  will  touch  the  a^axis,  as 
at  D',  and  the  two  corresponding  elbows  of  the  integral  curve  will 
coincide  as  shown  at  D.  Hence  the  slope  of  LM  has  the  same  sign 
on  both  sides  of  D.  The  integral  curve  therefore  changes  the  direc- 
tion of  its  curvature  at  D,  and  crosses  its  own  tangent,  which  it  cuts 
in  three  coincident  points.     Such  a  point  is  called  a  Point  of  Inflection. 

Ex.     Find  the  coordinates  of  the  elbows  of  the  following  loci : 

1.    y  =  x^-12x.  2.    2/ =  2x8 -16x2 +  24x4-5. 

3.    ?/ =  x3  -  6 x'-^  +  32.  4.    y  =  3x*- 20x8+ 18x2+ 108x. 

6.   y  =  3x6  -  20 x^^  +  10.  6.   y  =  3 X*  -  8 x8  -  66 x2  +  144x. 


98  THEORY  OF  EQUATIONS  [80 

Equal   Roots 

80.   Rolle's   Theorem.     At  least  one  real  root   of  the   equation 

f(x)=0  (1) 
lies  between  any  two  consecutive  real  roots  of 

f(x)  =  0.  ■               (2) 

For  there  is  at  least  one  elbow  of  the  integral  curve,  LM  (§  79), 
between  any  two  consecutive  intersections  of  it  with  the  a;-axis. 

Conversely,  LM  cannot  meet  the  ic-axis  more  than  once  between 
any  two  of  its  consecutive  elbows. 

Therefore,  at  most  one  real  root  of  (2)  lies  between  any  two  con- 
secutive real  roots  of  (1). 

That  is,  the  real  roots  of  (1)  separate  those  of  (2). 

If  by  a  continuous  modification  of  the  form  oif{x)  —  for  example, 
by  the  addition  or  subtraction  of  a  constant  (§  74)  —  two  roots  are 
made  equal,  the  root  of  f{x)  lying  between  them  must  approach  the 
same  value.     Hence  a  double  root  of  (2)  is  also  a  root  of  (1). 

In  general,  if  f{x)  has  an  r-fold  root,  such  a  root  being  regarded 
as  due  to  the  coalescence  of  r  distinct  roots,  then  will  /'  {x)  have  an 
(r  —  l)-fold  root  due  to  the  coalescence  of  the  (r  —  1)  intervening 
roots.  That  is,  if  f{x)  has  r  roots  each  equal  to  o,  f{x)  will  have 
(r  —  1)  roots  each  equal  to  a. 

Then,  by  the  application  of  Rolle's  theorem  to  f'(x)  and  f'\x), 
f"{x)  and  f"'(x),  and  so  on, 

if  f(x)  =  {x-  aycf>(x), 

we  have  .  f'(x)  =  (x  —  a)''~Vi(^)j 

f"(x)=^(x-ay-'<l>,(x), 


(3) 


j(r-i)(^)  =  (a:  -  a)cf>,_,(x). 

Conversely,  if  r  roots  of /'(a?)  coalesce  and  become  equal  to  a,  the 
corresponding  r  elbows  of  the  integral  curve  LM  will  coalesce ; 
then,  if  a  is  a  root  of  f(x),  this  r-fold  elbow  will  rest  on  the  «-axis 
and  give  an  (r  -j-  l)-f old  root  of  f(x). 

Hence,  if 

/-='(«)  =/"-''(«)  =/"-''(«)  =  •••/;(«)  =/(a)  =  0, 


80]  THEORY  OF  EQUATIONS  99 

and  a  is  a  single  root  of  p''~\x),  then  a  is  a  double  root  of  P''~^\x), 
a  triple  root  oi p''~^\x)y  •••  an  (r  — l)-fold  root  of /'(aj),  and  an  r-fold 
root  oif{x). 

This  suggests  an  easy  method  of  finding  real  multiple  roots  of  an 
equation,  when  the  roots  are  all  equal  except  one  or  two.  ^__ZI 

E.g.  it  /(ic)  =  x5-5a^  +  40a;2-80x  +  48  =  0, 

we  have  f'{x)  =  5  a;*  -  20  x^  +  80  a:  -  80, 

/"(a;)  =  20x3 -60x2  4- 80, 
/'"(x)  =  60x2-  120  X. 

The  roots  of  60  x^  -  120  x  =  0  are  0  and  2. 

Since/'"(2)  =  /"(2)  =  /'(2)=/(2)=  0,  2  isafourfold  root  of/(x)  =  0.  Hence 
all  its  roots  are  2,  2,  2,  2,  —  3. 

Moreover,  equations  (3)  are  true  whether  a  is  real  or  imaginary. 
For  suppose  f{x)  has  an  r-fold  root  equal  to  a,  then,  whether  a  is 
real  or  imaginary,  we  have  {%^^  and  §  67) 

f{x)  =  (x-aY<j>{x).  .  (4) 

In  this  case  the  given  function  f(x)  is  expressed  as  the  product  of 
two  distinct  functions  of  x,  viz.  {x  —  aj  and  <fi{x).  Hence  its  deriva- 
tive may  be  found  by  formula  (3),  §  60. 

That  is,    f{x)  =  (x-  ay  •  D,  [<^(x)]  +  <f>(x)  •  D,{x  -  ay.  (5) 

But       D,{x-ay=r{x-ay-^  •  DXx-a)  =  r{x-ay-^ .  [(7),  §  60]  (6) 

.-.  fix)  =  {x-  ay<t^'(x)  +  r(x  -  ay-'<t>(x)  (7) 

=  (a,  _  ay-'[{x  -  a)4>\x)  +  r<^(x)]  (8) 

=  {x-ay-'<i>,{x).  (9) 

That  is,  if  a  is  an  r-fold  root  oi  f{x),  then  it  is  also  an  (r—  l)-fold 
root  of  fix),  whether  a  is  real  or  imaginary. 

In  like  manner  if  f{x)  also  has  a  g-fold  root  equal  to  h,  and  an 
s-fold  root  equal  to  c,  and  so  on,  then 

f{x)  =  {x-ay{x-hyix-cy  ...  <^(^);  (10) 

and  f{x)  =  {x-ay-\x-hy-\x-cy'^  ...  4>,{x).  (11) 

...  (^x-ay-\x-hy-\x-cy-^'^* 

isthe  G.  C.  D  of  /{x)  and/'(x). 


100  THEORY  OF  EQUATION'S  [80 

Hence  the  multiple  roots  of  an  equation  f{x)  =  0,  if  there  are  any, 
can  be  detected  by  finding  the  G.  C.  D.  of  f(x)  and  f'(x)  by  the  usual 
algebraic  process. 

Likewise  the  common  roots  of  any  two  functions  can  be  obtained 
by  finding  the  G.  C.  D.  of  the  two  functions,  and  then  finding  the 
roots  of  this  G.  C.  D. 

Ex.     If  f(x)  =  x^  +  x^-lSx^-x'^-\-^8x-S6=0, 

then  /'(x)  =  5x*  +  4x3- 39x2-2£c  +  48. 

The  G.  C.  D.  oif(x)  and/'(ic)  will  be  found  to  be 

x^  +  X  -  6  =  (x  -  2)(x  +  S).    .'.  f{x)={x  -  2)2(x  +  3)2(x  -1)  =  0, 
and  the  roots  are  2,  2,   —  3,   —  3,  1. 


EXAMPLES 

Solve  the  following  equations  by  testing  for  equal  roots : 

1.  x^-\-  Ilx2  +  24cc-36zr0. 

2.  ic3-2ic2-15x  +  36r=0. 

3.  a;*-7a;3  + 9x2 +  27^-54  =  0. 

4.  X*- 11x3 +  44x2- 76x  + 48  =  0. 
6.  x*  -  5x3  -  9x2  +  81 X  -  108  =  0. 

6.  x5  -  15x3  +  10x2  +  60x  -  72  =  0. 

7.  x»  -  x*  -  5 x3  +  x2  +  8x  +  4  =  0. 

8.  x*  -  2x8  -  11x2 +12X  + 36  =  0. 

9.  x5- 10x2 +  15x- 6  =  0. 

10.  X* -3x3 -6x2  + 28x -24  =  0. 

11.  x5- 10x3 +  20x2- 15x  +  4=0. 

12.  X*  +  10x3  +  24x2 -32x- 128  =  0. 

13.  x5  +  19x4  +  130x3  +  350x2  +  125x  -  625  =  0. 

14.  x6  -  5x5  +  5x*  +  9x3  -  14x2- 4x  + 8  =  0. 

15.  x5-2x4-6x3  +  8x2  +  9x+ 2  =  0. 

16.  x6  +  7  x5  +  4x4  -  58x3  -  115x2  -  49x  -6  =  0. 

17.  x^^  -  8x3  +  24x2 -28x+ 16  =  0. 

18.  x5  -  6x3  -  28x2 -39x- 36  =  0. 

19.  What  is  the  condition  that  the  cubic  equation  x3  +  ^x  +  r  =  0  shall  have 
a  double  root  ? 

20.  Show  that  in  any  cubic  equation  with  rational  coefficients  a  multiple  root 
must  be  rational. 


81] 


QUADRATURE 


101 


Quadrature 

81.   Let  y  =  f(x)  and  yz=f'(x)  be  the  equations  of  the  continuous 
curves  LM  and  L'M'  respectively. 


It  is  required  to  find  the  area  included  between  the  curve  L'M', 
the  avaxis,  and  the  ordinates  corresponding  to  x  —  a  =  OQ,  and 
x  =  b=  OR,  where  b>a.     Let  K  denote  the  area  QA'B'R. 

Divide  the  distance  QR  into  (n  + 1)  equal  parts,  each  equal  to 
h  =  Sx.  Then  (n-\-l)h  =  b —  a.  Draw  ordinates  at  the  points  of 
division  and  construct  rectangles  as  shown  in  the  figure. 

Let      Xi  =  a-\-h=  OQi,     X2  =  a  +  2h,'-'Xn  =  a-\-nh=  OQn- 
Then  QA'=f'{a),     Q,  P/=/'(a^,)r-  QnPJ  =  f'(^nl 

and  the  sum  of  the  areas  of  the  (m  + 1)  rectangles  is 
hf'(a)  +  hfXx,)+hf'ix,)  +  -  hf'(x„). 


Now  f(x)  =  ^f/J^±R:iJM.     (§68.) 

Let  /-(.)+p  =  -^^^±4=M, 


where  p  is  a  quantity  that  approaches  zero  when  A  =  0. 
Then  lif{x)  +  lip  =  f{*  +  h)-f{x). 


(1) 
(2) 

(3) 
(4) 


102  QUADRATURE  [81 

Hence  we  may  put  /i/'(a)  +  hpQ  =  f(xi)  —/(a), 

¥'(^2)  +  hp2  =  f(x^  -  /(a-s), 


¥'(^n-l)  +  JiPn-1  =f(Xn)  -  f(.^n-l), 

From  these  equations  we  have  by  addition 

2hf(x)  +  ^hp=f{b)-f(a).  (5) 

The  second  member  of  (5)  is  independent  of  71,  %hf'(x)  represents 
the  sum  of  the  areas  of  the  (n  + 1)  rectangles  however  great  their 
number,  and  '^hp  =  0  when  h  =  0,  i.e.  when  n  becomes  infinite.  For 
"^hp  <  (n  -[-  l)hp'  =  (b  —  a)p',  where  p'  is  the  greatest  of  the  quanti- 
ties pi,  p2"-pn}  and  p'=  0  when  k  =  0. 

.•.^=,/r^  :^f'(x)8x=f(b)-f(a)  =  EB-QA.  (6) 

The  notation  used  to  express  this  is 

K=rfix)dx  =  f{b)-f(a),  (7) 

where  the  symbol  J  stands  for  ^Hhe  limit  of  the  sum,"  in  this  case, 
the  limit  of  the  smn  of  an  infinite  number  of  infinitesimal  rectangles. 

Therefore,  in  order  to  find  the  required  area,  we  must  first  obtain 
a  function  which  when  differentiated  will  give  f'(x) ;  then  substitute  in 
this  new  function /(x)  the  abscissas  of  the  bounding  ordinates  and 
take  the  difference  of  the  results.    Hence  equation  (7)  may  be  written 

In  applying  the  formula  we  must  first  find /(a;)  from /'(a;),  i.e.  we 
must  reverse  the  operation  of  differentiation.  In  this  sense  the  sym- 
bol r  denotes  an  operation  which  is  the  inverse  of  differ eyitiation. 

This  inverse  process  is  called  Integration. 

If  then  the  symbol  D  be  used  to  denote  differentiation,  the  two 

symbols  f  and  D  neutralize  each  other,  i.e.  \  Df(x)  =f{x). 


81] 


QUADRATURE 


103 


E.g.  if  Df{x)  =  f  {x)dx  =  (4  x^  -  3  x2  +  4  x  -  6)  dx,     . 

then  f -0/(35)  =  {f'{^)dx  =  f(x)  =  x*  -  x^  +  2  x^  -  6  x  +  c. 

Hence,  to  integrate  an  integral  function  of  x,  increase  tlie  exponent  of  each 
power  of  X  by  unity  and  divide  the  coefficient  by  the  increased  exponent. 

Thus,   I  x^cZx  = ,  provided  n  ^—1. 

J  n-\-l 

If  /'(it*)  is  the  derivative  of  f(x),  then  f(x)  is  called  the  Integral 
of  f'(x).  The  curve  LM  may  be  called  the  Integral  Curve  with 
respect  to  L'M'.  Then  we  may  say  that  the  area  bounded  by  the 
D.  C,  the  a>axis,  and  two  ordinates  is  numerically  equal  to  the  dif- 
ference of  the  two  corresponding  ordinates  of  the  I.  C. 

If  L'M'  lies  below  the  ic-axis  between  A'  and  B',  the  slope  of  LM 
between  A  and  B  will  be  negative  (§  79).  Hence  BB  <  QA,  i.e. 
f(h)  <f(a),  and  the  area  is  negative.  The  rectangles  will  then  lie 
above  the  curve. 

Therefore  the  area  will  be  positive  or  negative  according  as  it  lies 
to  the  right  or  left  of  the  curve  viewed  in  the  direction  of  x  increasing. 
If  L'M'  cuts  the  a>axis  between  A'  and  B',  the  formula  gives  the 
excess  (positive  or  negative)  of  the  area  which  lies  to  the  right  over 
that  which  lies  to  the  left. 

Ex.  1.  Find  the  area  of  the  segment  of 
the  parabola  y^  =  4ax  cut  off  by  the  double 
ordinate  through  P(x',  y'). 

Here  y  =  2  Vax"^  =f'(oc)- 

.-.  Area 

ONP  =  P  2  Vax^dx  =  2  y/aC  x^dx 


2V~a 


A^^'l- 


2  Va-|x 


/f 


=  1  x'.2\/ax'^  =  |xy 
=  f  rectangle  OBPN. 
.'.  Area  OPQ  =  f  rectangle  ABPQ. 

.*.  Area  between  AB  and   the   curve   is 
equal  to  ^  ABPQ. 

That  is,   the    parabola  trisects   the    rec- 
tanffle. 


B 

Y 

P 

^^^ 

x' 

y' 

O 

A 

N 

c 

^^^^ 

104 


QUADRATURE 


[82 


Ex.  2.     The  curve  y  =  x^  —  Sx^  -\- 2x  cuts  the  a-axis  in  the  points  (0,  0), 
B(h  0),  i>(2,  0). 

p  We  now  have  f'{x)  =x^  -Sx^  +  2x. 

:   OAB=  Cf'(x)dx 

=  Cix^  -Sx'^-\-2x)dx 
=  r  J  -  a:3  +  x2  +  cT  =  1. 

BCD  =z  r^  -  cc3  +  x2  +  cT 

=  (4-8  +  4  +  c)-a  +  c)=-i. 

|3 


-r-^         DEF  =  [j  -  x^ -{- x^ -{- cY 


DEF  =  ( 8^  -  27  +  9  +  c)  -  (4  -  8  +  4  +  c)  =  2f 


EXAMPLES 

1.  Find  the  area  included  between  the  curve  y  =  x^  —  Qx"^  ■\-2Zx  —  \b^  the 
X-axis,  and  the  lines  a;  =  1,  x  =  3 ;  also  x  =  3,  x  =  5;  a:  =  l,  5C  =  5. 

2.  Find  the  area  included  between  the  curve  ^  =  ^2  —  2x— 8,  the  x-axis,  and  the 
lines X  =  —  2,  x  =  4  ;  also  between  the  curve  y  =  x'^  —  2x  +  \  and  the  same  lines. 

Find  the  area  between  the  x-axis  and  the  curve 

3.  y  n:  x3  -  3 x2  -  9x  +  27.  4.   ?/  =  x^  +  ax^. 

5.    ?/  =  x*  -  4  x3  -  2  x2  +  12  X  +  9. 
Find  the  area  between  the  curves 

6.  ?/2  =:  4  ax  and  x^  =  4  ay.  9.    ?/"»  =  x"  and  y"  =  x"*. 

7.  ?/2  =  4  X  and  ?/2  =  x^.  10.   y  =  x^  —  x  and  y  =  x. 

8.  ?/3  =  x2  and  2/2  =  x^.  11.    y  =  x*  —  x  and  y2  =:  ^^^2. 


12 


ylws. 


Wl  —  71 


12.    y^  =  4  ax  and  y  =  2  x  —  4  a. 


-4ns. 


«'(^1 


13.  x2?/  =  a^,  x  =  b,  X  =  c,  and  ?/  =  0. 

14.  y  =  x2  —  5  X  +  4  and  x  +  y  =  4. 
16.  y  =  x^  and  ?/8  =  x. 

16.  Show  that  the  area  included  between  the  curve  y  =  Ax**,   the  x-axis 


and  the  line  x  =  a  is 


ah 


w  +  1 
Show  that  the  parabola  is  a  particular  case 


where  &  is  the  ordinate  corresponding  to  x  =  a. 


b2] 


MAXIMA  AND  MINIMA 


106 


Maxima  and  Minima 

82.   Let  the  curves  ML^  L'M\  and  L"M"  be  the  loci,  respectively, 
of  the  equations 

(1)  y=f(x),  (2)  y=f'(x),  (3)  y=f\x)r  ~ 

It  is  assumed  in  this  investigation  that  the  functions  f(x),  f'(x), 
f\x)  are  finite  and  continuous  for  all  finite  values  of  x. 
Then  7>"Jf "  is  the  Second  Derivative  Curve. 


Since  f\x)  is  the  first  derivative  of  f'ipi),  the  ordinate  of  L"M" 
at  any  point  represents  the  slope  of  L'M'  at  the  corresponding 
point;  and  the  intersections  E",  F'\  G"  of  X"J/"  with  the  a?-axis 
correspond  to  the  elbows  E',  F',  G'  of  L'M'  (§  79). 

Let  the  line  x  =  a  meet  the  curves  in  the  corresponding  points 
P,  P'j  P"f  and  the  x-axis  in  Q. 

Then         QP=:f(a),     QF  =f{a\     QP'^  =f"(a). 

That  is,  QP'  is  the  slope  of  LM  at  P,  and  QP"  is  the  slope  of 
L'M'  at  P. 

Suppose  the  point  P  to  move  along  the  curve  LM  toward  the  rigJU. 
As  P  approaches  the  elbow  B,  the  ordinate  QB  increases ;  but  as 
P  passes  through  B,  the  ordinate  ceases  to  increase  and  begins  to 


106  MAXIMA   AND  MINIMA  [82 

decrease.  At  such  a  point  the  ordinate,  i.e.  f(x),  is  said  to  have 
a  Maximum  Value,  or  to  be  a  Maximum.  In  like  manner  as  P 
approaches  the  elbow  A,  or  C,  the  ordinate  QP  decreases ;  but  as 
P  passes  through  A,  or  C,  the  ordinate  ceases  to  decrease  and  begins 
to  increase.  At  such  points  QP,  i.e.  f{x),  is  said  to  have  a  Minimum 
Value,  or  to  be  a  Minimum. 

That  is,  a  function,  fix),  is  said  to  have  a  maximum  value  when 
x  =  a,  if  /(a)  >/(a  ±  h) ;  and  a  minimum  value,  if  /(a)  <f(a  ±  h), 
for  very  small  values  of  li. 

Since  in  these  definitions  the  comparison  is  made  between  values 
of  f{x)  in  the  immediate  vicinity  only  of  A,  B,  C,  a  maximum  is 
not  necessarily  the  greatest,  nor  a  minimum  the  least,  of  all  the 
values  of  the  function. 

Moreover,  since  maximum  and  minimum  ordinates  occur  only  at 
the  elbows  of  a  curve  where  the  tangent  is  parallel  to  the  a?-axis, 
a  necessary  but  not  a  sufficient  condition  for  a  maximum  or  minimum 
value  of  fix)  is  f\x)  =  0  (§  79). 

Suppose  a  tangent  to  be  drawn  to  LM  at  any  elbow,  i.e.  at  any 
point  where  f  {x)  =0.  Then  the  curve  will  lie  below  or  above  this 
tangent  line  for  a  short  distance  on  both  sides  of  the  elbow,  accord- 
ing as  the  ordinate  of  the  elbow  is  a  maximum  or  a  minimum.  If 
the  curve  crosses  this  tangent,  as  at  D,  the  ordinate  is  neither  a 
maximum  nor  a  minimum. 

Hence,  as  P  passes  (toward  the  right)  through  an  elbow,  as  B, 
whose  ordinate  is  a  maximum,  the  slope  of  LM,  i.e.  f'(x),  changes 
from  positive  to  negative  ;  and  as  P  passes  through  an  elbow,  such 
as  A  or  C,  whose  ordinate  is  a  minimum /'(a?)  changes  from  negative 
to  positive. 

Therefore,  the  necessary  and  sufficient  conditions  that  f{x)  shall 
be  a  maximum  or  a  minimum  when  x  =  a  are  as  follows : 

For  max.,f'(a)  =0-f'(a  —  h),  positive;  f'{a  +  h),  negative,  i 
For  min.,  f  (a)  =0;  f  (a  —  h) ,  negative  ;  f  (a  +  h) ,  positive.    ) 

If  f'((^  +  '0  ^^^  /'(<*  —  ^)  have  the  same  sign,  /(a)  is  neither  a 
maximum  nor  a  minimum  value  of  f{x). 

Now  suppose,  as  is  usually  the  case,  that  «  is  a  single  root  of 
fix)  =  0,  so  that /"(a)  =^  0.     (§  80.) 


82]  MAXIMA  AND  MINIMA  107 

Then  if  QP  passes  through  a  maximum  value,  as  B'B,  when 
x  =  a,  the  slope  of  LM  changes  from  +  to  — .  Hence  the  corre- 
sponding point  P  crosses  the  a;-axis  from  above  dotonwards,  and  there- 
fore the  slope  of  L'M'  at  B'  is  negative,  i.e, 

B'B"  =f  "(a)  is  negative.  ~~^  ~ 

If  QP  passes  through  a  minimum  value,  as  C(7,  the  slope  of  LM 
changes  from  —  to  +.  Hence  P  crosses  the  avaxis  from  below 
upwards,  and   therefore  the  slope  of    L'M'  at  C  is  positive,  i.e. 

C'C"  =f"{a)  is  positive. 
Therefore,  if  /"  (a)  =f=  0,  the  necessary  and  sufficient  conditions 
that  f{a)  shall  be  a  maximum  or  a  minimum  value  of  f(x)  are : 

For  a  maximum,  f'(a)  =  0 ;  /"(ft),  negative.  |  ,-v 

For  a  minimum,  f'(a)  =  0 ;  f"((i),  positive.  > 

If  a  is  an  r-fold  root  of  f'(x)  =  0,  then  f"(a)  =  0  when  r>l 
(§  80)  and  the  conditions  (5)  fail  to  disclose  the  nature  of  the  corre- 
sponding ordinate. 

If  r  is  an  odd  number,  the  curve  L'M'  will  cut  the  avaxis  in  an 
odd  number  of  coincident  points,  and  hence  will  cross  the  a?-axis  at 
the  point  (a,  0).  Therefore  the  sign  of  /'(a;), will  change  from 
-f  to  —  for  a  maximum,  and  from  —  to  -f  for  a  minimum.  In 
this  case  we  must  use  conditions  (4)  to  determine  the  nature  of  /(a). 

If  r  is  an  even  number,  L'M'  will  not  cross  the  aj-axis  at  the  point 
(a,  0),  as  at  D'.  Hence  f'(x)  will  not  change  sign,  and  therefore 
/(a)  is  neither  a  maximum  nor  a  minimum. 

The  maximum  and  minimum  ordinates  of  L'M'  can  be  determined 
in  the  same  manner.  The  points  E,  F,  G,  D  on  IjM  corresponding 
to  the  maximum  and  minimum  ordinates  of  L'M'  are  therefore, 
respectively,  the  points  of  maximum  and  minimum  slope  of  LM. 
At  the  points  where  the  slope  of  a  curve  ceases  to  increase  and 
begins  to  decrease,  or  vice  versa,  the  curve  changes  the  direction  of 
its  curvature.  Therefore  E,  F,  G,  D  are  the  points  of  inflection 
of  LM  (§  79). 

Hence  the  position  of  the  points  of  inflection  of  a  curve  are  obtained 
by  finding  the  position  of  the  maximum  and  minimum  ordinates  of  the 

p.  a 


108 


MAXIMA  AND  MINIMA 


[83 


83.   Illustrative  Examples. 

Ex.  1.   The  curves  y  =  s'm  x  and  y  =  cos  x  are  good  examples  of  the  relations 
and  principles  explained  in  §  81  and  §  82. 

Let  /  (x)  =  sin  x. 

Then  fix)  =  ^%  ^n(x^h)-sinx  ^  ^lim^  ^^^^  ^^  ^  ,  ^^  sin|_^J        ^^^ 

=  cos  x.  (Ex.  13,  p.  71.)      (2) 

Similarly  it  has  been  shown  that  Dx(,cosx)  =  —  sin  x.     (p.  75.) 


Let 


and 


y  =f  (x)  =  sin  x,     equation  of  LM, 
y  =fi(x)  =  cos  X,   equation  of  L'M', 
y  =f"(x)  =  —  sin  x,  equation  of  L"M". 


Y 

A              p. 

-. 

^"^^^^""^^ 

/P 

X      ^ 

\./-'  / 

\/\ 

O             Q     \ 

/\                     \ 

\     \,- 

^^         y 

/       '--              \ 

.c..--\ 

-    L'  /\^__^ 

"■--»?:---' 

2   "  >    Z  "1   ^  "•) 


Then  /'(^)  =  cos  x  =  0,  when  x  =  ^  tt,  %ir, 

and  /"(I  tt)  =  —  sin  ^  TT  =  —  1.  .-.   sin  ^  tt  =  1  is  a  max. 

/"(I  tt)  =  —  sin  f  TT  =  1.  .-.  sin  f  ir  =  —  1  is  a  min.,  etc. 

Also  f"(^)  =  —  sin  X  =  0,  when  x  =  0,  tt,  2  tt,  3  tt,  etc. 

These  values  of  x  make  cos  x  alternately  a  maximum  and  a  minimum,  and 
hence  give  the  points  of  inflection  of  LM.  That  is,  the  sine  curve  changes  the 
direction  of  its  curvature  as  it  crosses  the  x-axis. 

Let  X  =  0^  be  any  line  parallel  to  the  y-axis. 

Then  f '(x)  =  cos  x=:  QF' =  slope  of  L3I  at  P. 

Moreover,  by  §  81  we  have 


Area  OAP' Q=  ( ''f'(x)dx  =  T ""cos  xdx  =  fsin xT  =  sin  x  =  QP. 


(3) 


That  is,  the  ordinate  of  any  point  of  the  cosine  curve  is  equal  to  the  slope 
of  the  sine  curve  at  the  corresponding  point;  and  the  ordinate  of  the  sine 
curve  is  equal  to  the  area  bounded  by  the  ordinate,  the  cosine  curve,  and  the 
axes  of  coordinates. 


83] 


MAXIMA   AND  MINIMA 


109 


Ex.  2.     Find    the    maximum    and 
minimum  values  of  the  function 

fix)  =  x4  -  4  x8  -  2  x2  +  12  X  +  4. 

Here  /'(x)  =  4  x^  -  12  x^  -  4  x  +  12 

and        /"(x)  =  12  x2  -  24  X  -  4. 

The  roots  of /'(x)  =0 

are  -1,   1,   3. 

/"(-I)  =32. 

/"(I)  =-16." 

/"(3)=32. 


/.  /(—  1)  =  —  5  is  a  minimum. 
.-.  /(I)  =  11  is  a  maximum, 
•*•  /(3)  =  —  5  is  a  minimum. 


The  roots  of  /"(x)  =  0  are  1  ±  f  ^3,- which  are  the  distances  of  the  points 
of  inflection  from  the  ^/-axis. 

In  the  solution  of  problems  in  maxima  and  minima,  we  must  first  obtain 
an  algebraic  expression,  /(x),  for  the  quantity  whose  maximum  or  minimum  is 
required.     We  may  then  proceed  as  in  the  preceding  examples. 

Ex.  3,  Find  the  maximum  rectangle  that  can  he  inscribed  in  a  given 
triangle. 

Let  h  =  the  base  of  the  given 
triangle  ABC,  h  the  altitude,  and  x 
the  altitude  of  the  inscribed  rectan- 
gle.   Then  from  similar  triangles, 

FG:b  =  ih-x):h. 

,:EG  =  \ih-x). 
h 

Then  -  {hx  —  x^)  is  the  area  of 
h 

the  rectangle,  which  is  to  be  made 

a  maximum.     Any  value  of  x  that 

will   make   (^x  —  x^)   a  maximum  will   also  make    -Qix  —  X')   a  maximum. 
Hence  we  may  put 

Then 
Also 


/(x)  =hx-  x2. 
/'(x)  =  /i  -  2  X  =  0  when  x  =  \h. 


/"(x)=-2. 
•*•  f{\^)  =  4  ^^  ^s  ^  maximum. 

Therefore  the  altitude  of  the  maximum  inscribed  rectangle  is  one-half  the 
altitude  of  the  triangle. 


110 


MAXIMA   AND   MINIMA 


[83 


Ex.  4.    Find  the  area  of  the  largest  rectangle  ivhich  can  be  inscribed  in  the 

ellipse 

Y  •        a;2     y2 


+  ^  =  1. 


(1) 


Let  K  denote  the  area  of  the 
rectangle.     Then 

K=2X'2y='^  Va^x^  -  x*    (2) 
a 

is  the  function  of  x  which  is  to  be 
a  maximum. 

Any  value  of  x  which  will  make 
a^x^  —  a;*  a  maximum,  or  a  mini- 
mum, will  also  make  K  a  maximum, 
or  a  minimum. 


Therefore,  let  f(x)  =  a^x^  -  a;*. 

Then  f<{%)  =  2  a^a;  -  4  x^  =  o  when  x  =  0,  or  ±  ^  a  V^, 

and  /"(x)  =  2  a^  -  12  x2  =  -  4  a2  when  x  =  ^  a  ^2. 

.\x  —  \a y/2  will  make  K  a  maximum. 

Therefore  K=2ab  is  the  area  of  the  maximum  rectangle,  which  is  half 
the  rectangle  whose  sides  are  the  axes  of  the  ellipse. 

Ex.  5.  Find  the  dimensions  of  a  cone  of  revolution  which  shall  have  the 
greatest  volume  with  a  given  surface. 

Let  X  =  the  radius  of  the  base,  y  =  the  slant  height,  V  =  the  volume,  and 
/S'  =  the  total  surface. 


Then 
and 


S  =  7rx2  +  TTxy ;  whence  y  =  —  —  x, 

TTX 

(Altitudey  =  ?/2  _  a;2  =  -^  _  2^. 

7r2x2  IT 


VS^x^  -  2  ttSx* 


Let 
Then 


.    y^^7rx2     r^2        2S^ 

3      >'7r2x2  T 

fix)  =  Sx'^-2  TTX*. 

/'(x)  =  2  6^x  -  8  7rx3  =  0  when  x  =  0,  or  ±  -  J-, 

'  /"(x)  =  2  19  -  24  7rx2  =  -  4  aS'  when  x  =  -J^-      • 

2  \7r 

.-.  F  is  a  max.  when  x  =  -^/-,  and  y  =  --./-• 

That  is,  if  the  surface  is  constant,  the  volume  of  the  cone  is  a  maximum 
•when  the  slant  height  is  three  times  the  radius  of  the  base. 


and 


83]  MAXIMA   AND  MINIMA  111 

EXAMPLES 

Find  the  maximum  and  minimum  ordinates  and  the  points  of  inflection 
(points  of  maximum  or  minimum  slope)  of  the  curves 

1.    y  =  x3  -  3  ic'^  +  4.  2.   y  =  x^- 9x^  +  15  x-S.       ^^  _ 

3.   y  =  ic8  -  3  x2  +  6  a;  +  7.  4.   y  =  x^  -  9  x^  +  2i  x  +  16. 

5.  Find  the  sides  of  the  maximum  rectangle  which  can  be  inscribed  in 
a  circle  ;  in  a  semicircle. 

6.  Find  the  sides  of  the  maximum  rectangle  which  can  be  inscribed  in  a 
semi-ellipse. 

7.  Find  the  altitude  of  the  maximum  rectangle  which  can  be  inscribed  in 
a  segment  of  a  parabola,  the  base  of  the  segment  being  perpendicular  to  the 
axis  of  the  parabola. 

8.  What  is  the  least  square  that  can  be  inscribed  in  a  given  square  ? 

9.  Find  the  altitude  of  a  cylinder  inscribed  in  a  cone  when  the  volume  of 
the  cylinder  is  a  maximum. 

la  What  are  the  most  economical  proportions  for  a  cylindrical  tin  can  ? 
That  is,  what  should  be  the  ratio  of  the  height  to  the  radius  of  the  base  that  the 
capacity  shall  be  a  maximum  for  a  given  amount  of  tin  ? 

11.  What  are  the  most  economical  proportions  for  a  cylindrical  tin  cup  ? 

12.  What  are  the  most  economical  proportions  for  an  open  cylindrical  water 
tank  made  of  iron  plates,  if  the  cost  of  the  sides  per  square  foot  is  two-thirds  of 
the  cost  of  the  bottom  per  square  foot  ? 

13.  An  open  box  is  to  be  made  from  a  sheet  of  pasteboard  12  inches  square 
by  cutting  equal  squares  from  the  four  corners  and  bending  up  the  sides.  What 
are  the  dimensions  of  the  largest  box  that  can  be  made  ? 

14.  If  a  rectangular  piece  of  pasteboard,  whose  sides  are  a  and  6,  have  a 
square  cut  from  each  corner,  find  the  side  of  the  square  so  that  the  remainder 
may  form  a  box  of  maximum  capacity. 

15.  A  person  being  in  a  boat  3  miles  from  the  nearest  point  of  the  shore, 
wishes  to  reach  in  the  shortest  possible  time  a  place  5  miles  from  that  point 
along  the  shore ;  supposing  he  can  walk  5  miles  an  hour,  but  can  row  only  at 
the  rate  of  4  miles  an  hour,  required  the  place  where  he  must  land. 

16.  The  cost  per  hour  of  driving  a  steamer  through  still  water  varies  as  the 
cube  of  its  speed.  At  what  rate  should  it  be  run  to  make  a  trip  against  a  four- 
mile  current  most  economically  ? 

1^7.  Find  the  altitude  of  the  greatest  cylinder  that  can  be  cut  out  of  a  given 
sphere. 


112  MAXIMA   AND  MINIMA  [83 

18.  Find  the  altitude  of  the  maxim uin  isosceles  triangle  that  can  be  inscribed 
in  a  given  circle. 

19.  Find  the  altitude  of  the  greatest  cone  that  can  be  inscribed  in  a  given 
sphere. 

20.  Find  the  altitude  of  a  cone  inscribed  in  a  sphere  which  shall  make  the 
convex  surface  of  the  cone  a  maximum. 

21.  If  the  slant  height  of  a  cone  is  constant,  what  is  the  ratio  of  the  radius 
of  the  base  to  the  altitude  when  the  volume  of  the  cone  is  a  maximum  ? 

22.  Find  the  dimensions  of  a  cone  with  a  given  convex  surface  and  a 
maximum  volume. 

23.  Find  the  altitude  of  the  least  cone  that  can  be  circumscribed  about  a 
given  sphere. 

24.  Find  the  altitude  of  the  maximum  cylinder  that  can  be  inscribed  in 
a  given  paraboloid. 

25.  What  is  the  diameter  of  a  ball  which,  being  let  fall  into  a  conical  glass 
of  water,  shall  expel  the  most  water  possible  from  the  glass ;  the  depth  of  the 
glass  being  6  inches  and  its  diameter  at  the  top  5  inches  ?  Ans.  m  in. 

26.  The  sides  of  a  rectangle  are  a  and  b.  Show  that  the  greatest  rectangle 
that  can  be  drawn  so  as  to  have  its  sides  passing  through  the  corners  of  the 

given  rectangle  is  a  square  whose  side  is  ^-X_2. 

27.  The  strength  of  a  beam  of  rectangular  cross-section,  if  supported  at  the 
ends  and  loaded  in  the  middle,  varies  as  the  product  of  the  breadth  of  the  cross- 
section  by  the  square  of  its  depth.  Find  the  dimensions  of  the  cross-section  of 
the  strongest  beam  that  can  be  cut  from  a  log  18  inches  in  diameter. 

28.  A  Norman  window  consists  of  a  rectangle  surmounted  by  a  semicircle. 
If  the  perimeter  of  the  window  is  given,  show  that  the  quantity  of  light  admitted 
is  a  maximum  when  the  radius  of  the  semicircle  is  equal  to  the  height  of  the 
rectangle. 

29.  What  are  the  most  economical  proportions  for  a  cylindrical  tin  can,  and 
a  cylindrical  tin  cup,  if  the  top  and  bottom  are  cut  out  of  regular  hexagons, 
and  allowance  is  made  for  waste  ?  Ans.  tt^  =  4  r  V  3,  and  irh  =2  r  VS. 

30.  Show  that  the  curve  (x^  +  a^)y  =  a^x  has  three  points  of  inflection,  and 
that  they  all  lie  on  the  line  x  =  4.y, 


97^ 


V{^ 


CHAPTER   VII 
CONIC  SECTIONS 

84.  The  general  equation  of  the  first  degree  and  also  some  special 
cases  of  the  equation  of  the  second  degree  have  been  considered  in 
Chapters  II  and  III.  We  now  proceed  to  the  general  equation  of  the 
second  degree,  and  the  standard  forms  to  which  it  can  be  transformed. 
It  will  presently  be  shown  that  the  locus  of  such  an  equation  is  al- 
ways a  curve  that  can  be  obtained  by  making  a  plane  section  of  a  right 
circular  cone.     For  this  reason  the  locus  is  called  a  Conic  Section.* 

85.  The  Fandamental  Property  of  a  Plane  Section  of  a  Eight  Cir- 
cular Cone,  or  a  Conic  Section. 

Let  VO  be  the  axis  of  a  right  circular  cone,  and  APB  any  section 
made  by  a  plane  not  passing  through  F. 

Inscribe  a  sphere  in  the  cone  tangent  to  the  plane  of  the  section 
at  F'j  then  the  line  of  contact  HRK  of  the  sphere  and  cone  is  a 
circle  with  centre  C  in  VO,  whose  plane  is  perpendicular  to  VO  and 
meets  the  plane  of  the  section  APB  in  the  line  ES. 

Pass  the  plane  FJO^  through  VO  perpendicular  to  the  plane  APB, 
meeting  it  in  the  line  AB,  meeting  the  plane  HKR  in  HK,  and  the 
line  ES  in  Z>;  then  the  plane  VMN  is  also  perpendicular  to  the 
plane  HKR,  and  therefore  perpendicular  to  ^*S'. 

*  After  studying  the  straight  line  and  the  circle,  the  old  Greek  mathematicians 
turned  their  attention  to  the  conic  sections,  and  by  investigating  them  as  sections  of 
a  cone  soon  discovered  many  of  their  characteristic  properties.  The  most  important 
of  these  discoveries  were  probably  made  by  Archimedes  and.Apollonius,  as  the  latter 
wrote  a  treatise  on  conic  sections  abovit  200  b.c. 

These  curves  are  worthy  of  careful  study,  not  only  on  account  of  their  historic 
interest,  but  also  on  account  of  their  importance  in  the  physical  sciences  and  their 
frequent  occurrence  in  the  experiences  of  everyday  life.  For  example,  the  orbit  of  a 
heavenly  body  is  a  conic  section.  For  this  reason  they  were  thoroughly  studied  by 
the  astronomer,  Kepler,  about  1600  a.d.  The  path  of  a  projectile  is  a  parabola. 
The  graphical  representations  of  the  law  of  falling  bodies,  the  pressure-volume  law 
of  gases,  the  law  of  moments  in  uniformly  loaded  beams,  are  all  conic  sections.  The 
bounding  line  of  a  beam  of  uniform  strength,  the  oblique  section  of  a  stove-pipe,  the 
shadow  of  a  circle,  the  apparent  line  dividing  the  dark  and  light  parts  of  the  moon, 
etc.,  are  conic  sections.    The  reflectors  in  head-lights  and  search-lights  are  parabolic. 

113 


114 


CONIC   SECTIONS 


E 


[85 


N 


I 

^ 


85]  CONIC   SECTIONS  115 

Let  P  be  any  point  on  the  section. 

Draw  PF,  and  the  element  PV  which  will  be  tangent  to  the 
sphere  at  R. 

Through  P  draw  a  line  perpendicular  to  the  plane  HKRy  which 
will  meet  CR  produced  in  Q ;  and  through  PQ  pass  a  plane  perpen^ 
dicular  to  ES  meeting  it  in  S. 

Let  /3  =  Z  PRQ  =  Z  AHD,  the  complement  of  the  semi-vertical 
angle  of  the  cone.  Let  a  =  Z  ADH=  Z  PSQ.  Then,  since  tangents 
from  an  external  point  to  a  sphere  are  equal,  PF=  PR. 

From  the  right  triangles  PQR  and  PSQ  we  get 

Pq  =  PR^\np  =  PS^ina.  p  $ 

.   PF_sin«  f/  .      -^)     ... 

••:^-^i^*        i..4     ^'''  ^^^ 

So  long  as  we  consider  any  particular  section,  the  point  F  and  the 
line  ES  are  fixed,  a  is  constant,  and  therefore  the  ratio  of  PF  to  PS 
is  constant. 

Equation  (1)  expresses  the  Fundamental  Property  of  a  Conic  Sec- 
tion, which  is  used  as  the  defining  property.  Moreover,  all  curves 
which  have  this  property  are  plane  sections  of  some  cone ;  for  all 
possible  curves  satisfying  this  condition  are  gotten  by  giving  this 
constant  ratio  all  possible  values,  and  also  letting  the  distance,  FD, 
from  the  fixed  point  to  the  fixed  line  have  all  possible  values.  We 
can  do  this  with  a  conic  section.  For  any  particular  value  of  )8,  i.e. 
for  any  particular  cone,  the  ratio  can  vary  from  zero  (when  a  =  0)  to 
CSC  (3  (when  «  =  ^  tt).  For  any  particular  value  of  a  the  ratio  can 
vary  from  sin  a  (when  P  =  \tt)  to  oo  (wheu  (i  =  0).  Thus  the  ratio 
can  have  any  value  from  0  to  oo .  Also  the  distance  of  F  frora  ES, 
depending  as  it  does  upon  the  size  of  the  inscribed  sphere,  for  any 
particular  cone  and  any  particular  value  of  a  can  vary  from  zero 
to  oo  .  Therefore  the  property  expressed  by  (1)  is  indeed  a  defining 
property  of  a  conic  section,  that  is  : 

A  Conic  Section,  or  a  Conic,  is  the  locus  of  a  point  which  moves  in  a 
plane  so  that  its  distance  from  a  fixed  point  in  the  plane  is  in  a  con- 
stant ratio  to  its  distance  from  a  fixed  line  in  the  plane.* 
• 

*  This  is  generally  known  as  Boscovich's  definition  of  a  conic  section,  bnt,  in  the 
article  on  Analytic  Geometry  in  the  Encyclopedia  Britannica,  nintli  edition,  Cay  ley 
calls  it  the  definition  of  ApoUonius. 


116  CONIC   SECTIONS  [8f- 

The  fixed  point  F is  called  the  Focus;  the  fixed  line  ES  is  called 
the  Directrix;  the  constant  ratio  is  called  the  Eccentricity,  and  k 
denoted  by  the  letter  e ;  the  line  BFD,  through  the  focus  perpen- 
dicular to  the  directrix,  is  called  the  Principal  Axis  of  the  conic. 

86.    Classification  of  the  Conic  Sections. 

Using  e  to  denote  the  eccentricity,  we  have,  by  (1)  of  §  85, 

PF     sin  a  z^n 

When  a<p,  e<l;  the  plane  of  the  section  meets  all  the  ele- 
ments of  the  cone  on  the  same  side  of  the  vertex ;  the  section  is  a 
closed  curve  as  shown  in  the  figure  §  85,  and  is  called  an  Ellipse. 

When  a  =  0,  e  =  0 ;  the  plane  of  the  section  is  perpendicular  to 
the  axis  of  the  cone,  VO,  and  the  section  is  a  Circle.  Hence  a 
circle  is  a  particular  case  of  the  ellipse. 

When  «  =  ^,  6  =  1 ;  the  line  AB  (§  85)  is  then  parallel  to  VN, 
and  the  point  B  moves  off  to  an  infinite  distance ;  the  section 
consists  of  a  single  infinite  branch,  and  is  called  a  Parabola. 

When  a>  ^,  e  >  1,  and  the  plane  APB  (§  85)  meets  NV  produced 
on  the  other  sheet  of  the  conical  surface ;  the  section  is  then  com- 
posed of  two  infinite  branches,  one  lying  on  each  sheet  of  the  cone, 
and  is  called  a  Hyperbola. 

Thus  the  parabola  is  the  limiting  case  of  both  the  ellipse  and  the 
hyperbola. 

Let  the  plane  of  the  section  pass  through  the  vertex  of  the  cone. 

Then  if  e  <  1,  the  section  is  a  point  ellipse  or  a  point  circle. 

If  e  =  1,  the  plane  is  tangent  to  the  cone,  and  the  parabola  reduces 
to  two  coincident  straight  lines. 

If  e  >  1,  the  hyperbola  becomes  two  intersecting  straight  lines, 
which  approach  in  the  limit  two  parallel  lines  as  the  vertex  of  the 
cone  moves  off  to  an  infinite  distance. 

Hence  a  point,  two  intersecting  straight  lines,  two  parallel  straight 
lines,  and  two  coincident  straight  lines  are  all  limiting  cases  of  conic 
sections. 

Under  the  head  of  conic  sections  we  must  therefore  include : 

(1)  Tlie  Ellipse,  including  the  circle  and  the  point; 

(2)  The  Parabola;  (3)   Tlie  Hyperbola;  (4)   The  Line-pair. 


87] 


CONIC   SECTIONS 


117 


EXAMPLES 

1.  Inscribe  a  sphere*  tangent  to  the  plane  APB  (fig.  §  85)  on  the  other 
side  and  thus  show  that  the  ellipse  has  another  focus  and  a  corresponding 
directrix;  and  that  the  two  directrices  are  parallel  and  equidistant  from  the 
foci. 

2.  By  means  of  these  two  inscribed  spheres,  prove  the  property  of  the  ellipse 
given  in  §  34. 

3.  Inscribe  spheres*  in  both  sheets  of  the  cone  and  show  that  the  hyperbola 
also  has  two  foci  and  two  directrices. 

4.  Prove  the  property  of  the  hyperbola  stated  in  §  36. 

5.  Where  are  the  foci  and  the  directrices  of  the  circle,  the  parabola,  and 
two  intersecting  straight  lines  ? 


iX, 


General   Equation   of   the   Conic    Sections 
87.     To  find  the  equation  of  a  conic  section  in  rectangular  coordinates. 


I.   Let  the  equation  of  the  directrix  EC  be 

X  cos  a  -\-  y  sin  a  —  }}  =  0. 
Let  F(k,  I)  be  the  corresponding  focus. 
Let  P(x,  y)  be  any  point  on  the  conic. 
Draw  PS  perpendicular  to  EC,  and  join  P  and  F. 
Then  from  equation  (1)  of  §  86  we  have 

PF=e'PS. 


(1) 


(2) 


*  For  complete  diagrams  see  Some  Mathematical  Curves  and  their  Graphical 
Construction,  by  F.  N.  Willson,  pp.  45,  46.  Also  his  Descriptive  Geometry,  pp. 
44,  45.  - 


118  CONIC   SECTIONS  [87 

Now  PF''  =(x-  ky  4-  (2/  -  l)%  [(2),  §  7.] 

and  PS  =  xGOsa-\-y  sina^p.  [(4),  §  47.] 

Therefore  the  required  equation  is 

(x  —  ky  -\-  (y  —  ly  =  e^(x  cos  a -{-y  sin  a—  py.  (3) 

Expanding  (3)  and  collecting  terms  we  have 
(1  —  e^  cos^  a)a^  —  2{e^  sin  a  cos  a)xy  +  (1  —  e^  sin^  a)y^ 

+  2(e^p  cos  a  -  k)x  +  2{fp  sin  a  -  V)y -[■  1^ -\- 1"^  -  eY  =  0.  (4) 

Since  equation  (4)  contains  five  arbitrary  constants,  k,  Z,  a,  p,  e, 
it  may  be  any  equation  of  the  second  degree.  That  is,  any  equa- 
tion of  the  second  degree  represents  a  conic  section. 

The  most  general  equation  of  a  conic  is,  therefore,  the  complete 
equation  of  the  second  degree,  and  may  be  written 

ax'  +  2hxy  +  bf  +  2gx  +  2/^/  +  c  =  0.  (5) 

II.  Let  the  directrix  be  taken  as  the  y-Sixis,  the  principal  axis, 
FD,  (§  85)  as  the  o^-axis.  Then  a  =  l=p  =  0,  and  k  =  DF.  There- 
fore the  equation  of  the  conic  (3)  takes  the  simple  form. 

(x  —  A;)^  -\-y^  =  e'x^f 
or  (1  —  e^)a^  -\-y'  —  2kx 

If  a^  =  0  in  (6),  then  y  =  ±  kV^^. 

Hence  a  conic  does  not  intersect  its  directrix. 

If  2/  =  0,  then  there  are  two  real  values  of  x,  viz., 

Therefore  a  conic  section  cuts'  its  principal  axis  in  two  points. 
These  points  are  called  the  Vertices  of  the  conic.  The  point  mid- 
way between  the  vertices  is  called  the  Centre  of  the  Conic. 

The  Latus  Rectum  of  a  conic  is  the  chord  through  either  focus 
perpendicular  to  the  principal  axis. 

To  find  its  length,  \Qt  x  =  k  in  (6),  then 

y  =  ±  ekf  and  2y  =  Latus  Rectum  =  2 ek. 

The  different  cases  corresponding  to  the  different  values  of  e  will 
now  be  separately  considered. 


^'^'  I       X  (6) 


88] 


CONIC   SECTIONS 


119 


Standard  Equations  of  the  Conic  Sections 
88.   The  Parabola.     e  =  l. 

When  e  =  1,  equations  (7)  of  §  87  give 


a^  =  |A;  =  Z>0, 


072  =  - =00. 


Hence  the  parabola  has  one  vertex  midway  between  the  focus  and 
directrix,  and  the  other  at  infinity.* 


When  e  =  ly  equation  (6)  of  §  87  gives  for  the  equation  of  the 
parabola  referred  to  its  axis  and  directrix 

f  =  2k(x-i7c).  (1) 

Let  a  =  ik  =  DO=  OF-,  then  this  equation  becomes 

y^  =  4:a(x-a).  (2) 

Now  write  a;  +  a  in  the  place  of  x ;  this  moves  the  origin  to  the 
vertex  0(af  0)  [§  53,  (1)],  and  the  equation  becomes 

y^  =  ^ax,  (3) 

which  is  the  standard  form  of  the  equation  of  the  parabola. 
When  a;  =  a  in  (3),  y  =  ±2a. 

.-.   L'L  =  4  a  =  Latus  Rectum. 
Ex.  Construct  the  parabola,  having  given  the  focus  and  the  directrix. 


♦  Compare  this  result  with  the  position  of  B  in  the  figure  of  §  86  when  a  =  /S. 


120 


CONIC   SECTIONS 


[89 


89.   The  Ellipse.     e<l. 

When  c  <  1,  the  two  a;-intercepts  [(7),  §  85]  are  both  finite  and 
positive;  that  is, 

^1 


1  +  e 

k 
1-e 


=  DA'>k. 


Hence  the  ellipse  has  two  vertices  lying  on  the  same  side  of  the 
directrix,  but  on  opposite  sides  of  the  focus. 


c 

R 

B 

Y 

P 

R' 
X 

'-f 

L 

/> 

\^ 

D 

.A 

P 

O               Q    r          /A' 

D' 

^-               NJ 

|i;    ■■ 

B' 

y'^ 

Let  O  be  the  centre,  and  let  AA  =  2  a. 


Then 

whence 
Also 


2a  =  X2  —  Xi 


k 


^  2ek 

1-e      l+e     l-e" 

a  k 

e 
a 


1-e' 


k  = ae. 

e 


l,0  =  i(.,  +  ..)  =  i(^  +  ^) 


a 


1-e'     e 


.:   FO=DO-DF=~~k  =  ae. 


(1) 

(2) 

(3) 
(4) 


8»]  CONIC  SECTIONS  121 

Substituting  in  equation  (6)  of  §  85  the  value  of  k  given  by  (2) 
gives  for  the  equation  of  the  ellipse  referred  to  DC  and  DX 

(x---\-  ae\+  f  =  6^x2.  (5) 

The  origin  may  be  transferred  to  the  centre,  0  (  -,  0  J,  by  writing 
aj  +  -  in  the  place  of  x  [§  53,  (1)]  ;  this  gives 

(a;  +  ae)2  +  /  =  e2(^aj  +  fj, 
or  a^(l  -  e'-^  +  /  =  a'(l  -  e^- 

.-.  ^  + t =  1.  y^  (6) 

When  a;  =  0,  we  have 

2/  =  ±  a  Vl  —  e^ ; 

which  gives  the  ?/-intercepts  OB  and  OB?. 
If  these  lengths  are  denoted  by  ±  6,  we  have 

62=a2(l-e2),  (7) 

and  equation  (6)  takes  the  standard  form 

Since  e  <  1,  6  <  a  from  (7)  ;  therefore 

Hence  the  line  AA}  is  called  the  Major  Axis,  and  BB^  is  called  the 
Minor  Axis  of  the  ellipse. 

Take  OF'  =  FO  and  OD'  =  DO;  draw  D'C  perpendicular  to  OX. 
Then  F'  is  the  other  focus,  and  D'C  the  corresponding  directrix 
(Ex.  1,  p.  117).  Hence  the  foci  are  the  points  F'  (ae,  0)  and  F{—  ae,  0) 
from  (4)  ;  and  the  equations  of  directrices  are,  from  (3), 


a 


<»  =  ±~-  (9) 


e 


Let  P(x,  y)  be  any  point  on  the  ellipse ;  draw  a  line  through  P 
parallel  to  AA'  meeting  the  directrices  in  R  and  R\  and  draw  PQ 
perpendicular  to  AA'. 

*  For  a  discussion  of  this  equation  see  §  35. 


122  CONIC   SECTIONS  [89 

Then  FP^e-RP, 

and  F'P  =  e'B'P.  [(2),  §  87.] 

.-.   FP  =  e'DQ  =  e(DO+OQ) 

=  ef^-\-x)  =  a  +  ex,  (10) 

and  F'P=  e  •  QD'  =  e(OD' -  OQ) 

=  el xj  =  a  —  ex.  (11) 

Whence  FP  +  F'P  =2  a.  (Cf.  §  34.)     (12) 

From  equations  (7)  and  (4)  we  get 


ae  =  Va^  -  b-  =F0=  OF'. 

To  find  the  length  of  the  latus  rectum  we  put  a;  =  ±  ae  in  (8)  ;  this 
gives 

2/'  =  b\l-  e")  =  - .  from  (7) 

.:   X'i  =  ?|^.  (14) 

If  tt  =  6,  equation  (8)  reduces  to 

a;2  +  2/2  =  a^, 

and  equations  (13),  (4),  and  (3),  respectively,  give 

e  =  0,    FO=OF'=0,    DO=OJ)'  =  oo. 

That  is,  the  circle  is  the  limiting  form  of  the  ellipse,  as  the  eccen- 
tricity approaches  zero,  and  the  directrices  recede  to  infinity. 

Ex.    Construct  an  eUipse,  having  given  the  foci  and  the  length  of  the  major 


-  T      n        •  distance  between  foci       ,    ^,    ,.  ,  ,  .   «   .^    .    .i. 

*  In  all  comes  e  =  37-^ i— ; ^. — ;  both  distances  become  infinite  in  the 

distance  between  vertices 

parabola,  and  both  become  zero  in  the  case  of  two  intersecting  lines.  (See  also  (11),  §90.) 


90] 


CONIC   SECTIONS 


123 


90.   The  Hyperbola.     e>l. 

From  equations  (7)  of  §  87  we  have  for  the  vertices 


aji  = 


and  X2 


k 


1+e  1-e 

Since  e  >  1,  Xi=DA  <  k,  and  X2  =  DA'  is  negative. 
Therefore,  the  hyperbola  has  two  vertices  lying  on  the  same  side 
of  the  focus  but  on  opposite  sides  of  the  directrix. 


Let  0  be  the  centre,  and  let  A'A  =  2  a. 


Then 


2  a  =  A'D  +  DA  =  —  x2  +  Xi 
k      .      k  2ek 


e-1 
a         k 

e 


+ 


e  + 1      e'^  —  l 


e'-l 


and  k  =  ae  — 


30     - 


(1) 
(2) 


124  CONIC  SECTIONS  [90 

The  equation  of  the  hyperbola  referred  to  DC  and  DX  is,  from 
(2),  and  (6)  of  §  87, 

/aj-ae-h-Y  +  Z^eV.  (5) 

Moving  the  origin  to  the  centre  0(  —  -,  0  ]  gives 


(a;  —  aef  -\-y^  =  e'^[x i  > 


or  — H ^ =  1.  (6) 


Since  e  >  1,  the  quantity  a^(l  —  e-)  is  negative;  if  we  put 

-b^  =  a-(l-e-), 
or  52^^2(g2_-|^s^^  ^^^^ 

equation  (6)  reduces  to  the  standard  form 

^_^-l  (8) 

When  X  =  0,  y  =  ±bV—l.  Since  these  vahies  of  y  are  both 
imaginary,  the  hyperbola  does  not  meet  the  line  through  its  centre 
perpendicular  to  its  principal  axis  in  real  points ;  but,  if  B,  B'  are 
points  on  this  line  such  that  B'O  =  OB  =  b,  the  line  BB'  is  called 
the  Conjugate  Axis.  The  line  A  A'  joining  the  vertices  is  called  the 
Transverse  Axis. 

On  the  line  OX  take  OF'  =  FO,  and  OD'  =  DO ;  then  F'  is  the 
other  focus  and  D'C,  perpendicular  to  OX,  is  the  corresponding 
directrix  (Ex.  3,  p.  117).  Hence  the  coordinates  of  the  foci  are 
(  ±  ae,  0),  from  (4),  and  the  equations  of  the  directrices  are,  from  (3), 

oo  =  ±^'  (9) 

e 

As  in  the  ellipse,  we  find  the  latus  rectum 


LL'  =  ^^'  (10) 

a 


Equations  (7)  and  (4)  give 


ae=Va:'  +  b'=OF. 

_V^:^i  ^OF  ^F'F  _  ,j^. 

a  OA     A' A  ^     ^ 


91]  CONIC   SECTIONS  125 

Let  P(a7,  y)  be  any  point  on  the  hyperbola ;  draw  a  line  through  P 
parallel  to  AA^  meeting  the  directrices  in  R  and  R',  and  draw  PQ 
perpendicular  to  AA\ 

Then  FP^e-  RP,         F'P  =  e  •  WP.     [(2),  §  87.] 

.-.  FP^e'DQl=e{Oq-OB)=e{x-^  =  ex-a',  (12) 

andi<^'P=e.Z)'Q  =  e(Oe  +  i>'0)=e('aj  +  -)  =  ea;  +  a.  (13) 

Whence  F'P  -Fr  =  2a.  {Of.  §  36.)     (14) 

If  a  =  6,  the  equation  of  the  hyperbola  becomes 

x'^-y'^  =  a^.  (15) 

This  is  called  the  Equilateral  or  Rectangular  Hyperbola.      (See 
§§  169,  170.) 
Then  from  (11),  (3),  and  (4)  we  have,  respectively, 
e  =  V2,     OD  =  i  a  V2,     OF  =  a  v'2. 

Ex.  Construct  a  hyperbola,  having  given  the  foci  and  the  distance  between 
the  vertices. 

91.  Limiting  cases  of  conic  sections. 
If  A;  =  0,  equation  (6)  of  §  87  reduces  to 
2/2  =  arXe2-l). 

This  equation  represents  two  straight  linesy  which  are  real  if  e  >  1, 
coincident  if  e  =  1,  and  imaginary,  but  with  a  real  point  of  intersec- 
tion, if  e  <  1. 

From  (7)  of  §  87  we  then  have  a^j  =  x^  =  0.  Hence  the  foci,  the 
vertices,  and  the  centre  of  two  intersecting  lines  all  coincide  on  the 
directrix.     The  two  directrices  also  coincide. 

When  e=cc  (a  being  finite),  the  equation  of  the  hyperbola  [(8), 
§  90]  reduces  to  x^  =  a',  which  represents  two  parallel  lines.  Equa- 
tions (3)  and  (4)  of  §  90  then  show  that  the  foci  of  two  parallel  lines 
(considered  as  the  limiting  case  of  a  hyperbola)  are  at  infinity  while 
their  directrices  coincide  and  are  equidistant  from  the  two  lines. 

Hence  we  must  consider  two  intersecting  lines,  real  or  imaginary 
{i.e.  a  real  point),  two  coincident  lines,  and  two  parallel  lines  ns 
limiting  cases  of  conic  sections.     (Cf.  §  86.) 


Jv^V 


126  CONIC   SECTIONS  [92 

Tangents 

92.    To  find  the  equation  of  the  tangent  to  the  conic  represented  by  the 
general  equation 

ax'  +  2hxy  +  bf  +  2gx  +  2fy-hc  =  0.  (1) 

The  equation  of  the  tangent  to  any  curve  f(x,  y)  =  0  at  the  point 
(x',  y')  is  (§  62) 

2/-2/'  =  ||!(^-«^').  (2) 


For  equation  (1)  we  have  found  in  §  61 

dy  ^      ax-\-hy-{-g^ 
dx         hx  4-  by  -+-/ 

Therefore  the  required  equation  is 


(3) 


or  axx^  +  h  (xy'  +  x'y)  +  byy'  +  gx  +fy 

=  ax"  +  2hx'y'-\-by"  +  gx'+fy'.     (5) 

Add  gx'  -\-fy'  +  c  to  both  sides  of  (5) ;  then,  since  (x',  y')  is  on 
the  conic,  the  right  member  will  vanish  and  we  have  the  required 
equation, 

axx'  +  h  (xy'  +  x'y)  +  byy'  +  gr  (a?  +  x')  +fiy  +  y')+c  =  0,     (6) 

Observe  that  the  equation  of  the  tangent  at  (x',  y')  is  obtained 
from  the  equation  of  the  conic  by  writing  xx'  for  xr^  x'y  +  xy'  for  2  xy, 
yy'  for  /,  x  +  x'  for  2  x,  and  y  -{-y'  for  2  y.  Note  also  that  putting  x 
for  x'  and  y  for  y'  in  (6)  reproduces  the  equation  of  the  curve. 

E.g.  the  equation  of  the  tangent 

to  the  circle  x'^  +  y'^  =  r^      at  the  point  (x',  y')  is  xx'  +  yy'  =  r^, 

to  the  parabola  y^  =  4ax  at  the  point  (x',  y')  is  yy'  =  2a(x-{-  x'), 

to  the  ellipse        ^  + 1^  =  1       at  the  point  (x',  y')  is  ^  +  ^'  =  1, 
to  the  hyperbola  ^  _  |^  =  1       at  the  point  (x',  ?/')  is  ^  -  ^  =  1. 


/I'' 

93]  CONIC   SECTIONS  127 


Jsj^!) 


93.  Two  tangents  can  be  drawn  to  a  conic  from  any  pointy  which 
will  be  realf  coincident j  or  imaginary ^  according  as  the  point  is  outside, 
on,  or  within  the  curve. 

Let  the  equation  of  the  conic  be  [§  87,  (6)]  

aa^  +  f-{-2gx-{-g'  =  0,  (1) 

where  a  =  1  —  e^,  and  g=  —k. 

Let  (Jij  Tc)  be  any  point  j  then  the  equation  of  any  line  through 
this  point  will  be  (§  43) 

y  —  k  =  m(x  —  h).  (2) 

Eliminating  y  between  (1)  and  (2)  gives 

(a  +  m^x"  +  2(km  -  hm^  +  g)x  -f  ^V  -  2  hkm  -{-k^  +  g^  =  0.  (3) 

The  roots  of  (3)  are,  by  §  24,  the  abscissas  of  the  points  of  inter- 
section of  (1)  and  (2).  If  these  roots  are  equal,  the  points  of  in- 
tersection will  coincide  and,  by  §  57,  (2)  will  be  tangent  to  (1).  The 
condition  that  (3)  shall  have  equal  roots  *  is 

{km  -  hm^  -f  gf  =  (a  +,  m^){hV  -  2  hkm  +  k^  -[-g^  (4) 

or        {ah^  +  2  gh  -^g") m"  -  2{ahk  -f  gk)m  +  {ale"  +  ag''-g^  =  0.      (5) 

Equation  (5)  is  a  quadratic  in  m  whose  roots  are  the  slopes  of 
the  tangents  from  (/i,  k)  to  the  conic.  Since  a  quadratic  equation 
has  two  roots,  two  tangents  will  pass  through  any  point  (Ji,  k). 

The  conic  is,  therefore,  a  curve  of  the  second  class. 

The  roots  of  (5)  are  real,  equal,  or  imaginary,  according  as 

a/i2  4-A;2  +  2^/i  +  />,=,  or  <0.  (6) 

Therefore  the  tangents  are  real,  coincident,  or  imaginary  accord- 
ing as  the  point  (h,  k)  is  outside,  on,  or  within  the  conic.  (§  20,  II.) 
(The  directrix  is  outside,  the  focus  inside  the  conic.) 

Since  equation  (3)  is  a  quadratic  in  x,  any  straight  line  meets  a 
conic  in  two  points,  which  may  be  real,  coincident,  or  imaginary. 

Therefore  the  conic  is  also  a  curve  of  the  second  order. 

If  e  =  1  and  m  =  0,  then  a-\-  m^  =  0,  and  hence  one  root  of  (3) 
is  infinite  (§  77).  Therefore  a  straight  line  parallel  to  the  axis  of 
the  parabola  meets  the  curve  in  one  point  at  a  finite  distance,  and  in 
another  at  an  infinite  distance  from  the  directrix. 

*  The  two  roots  of  ax^  -f  6x  +  c  =  0  will  be  equal,  if  6^  =  4  ac. 

The  method  here  used  is  worthy  of  special  attention  because  of  its  wide  application. 


128 


CONIC   SECTIONS 


[94 


Pole  and  Polar 

94.   The  equation  of  the  tangent  to  the  conic 

ax'  +  y'-^2gx+g'  =  0  (1) 

at  the  point  (a;',  2/'),  if  this  point  is  07i  the  conic,  is  (§  92) 

axao'  +  yy'  +  g(.oo  +  oc')  +g^  =  0.  (2) 

Suppose,  however,  that  F'(x'j  y')  is  7iot  on  the  conic.  Then  what 
is  (2)?  It  still  has  a  meaning,  still  represents  a  straight  line  related 
in  a  definite  way  to  the  point  (x',  y')  and  the  conic  (1).  Moreover 
this  line  will  cut  the  conic  in  two  points  (§  93). 


Let  these  points  be  Pi(xi,  y^)  and  P^ix^,  2/2)- 
Then  the  equations  of  the  tangents  at  these  points  are  (§  92) 
axx^  +  yyi  4-  g(x  +  x^)  -\-g'^  =  0, 
and  axx2  -\-  yy^  -\-  g{x  -\-  ^2)  -\-  9^  =  0- 

The  conditions  that  (3)  and  (4)  shall  pass  through  {x\  y')  are 
ax%  +  2/ '2/1  +  9(:^'  +  ^i)  +f  =  0, 
and  ax%  +  y'y,  +  g{x'  +  x,)  +  /  =  0. 

But  (5)  and  (6)  are  also  the  conditions  that  (2)  shall  pass  through 
both  of  the  points  (xi,  y{)  and  (xo,  y^)- 

Therefore  (2)  is  the  line  passing  through  the  points  of  contact  of 
the  tangents  from  the  point  F'(x',  y'). 

The  point  (x',  y')  and  the  line  (2)  are  called  Pole  and  Polar  ivith 
respect  to  the  conic  (1). 


(3) 
(4) 

(5) 
(6) 


95]  CONIC   SECTIONS  129 

The  tangents  from  the  point  (x',  y^  will  be  real  or  imaginary 
according  as  (x',  y')  is  outside  or  inside  the  conic  (§  93) ;  but  the 
line  (2)  is  real  when  (x'j  y')  is  real.  So  that  there  is  always  a 
real  line  passing  through  the  imaginary  points  of  contact  of  the 
two  imaginary  tangents  drawn  from  a  point  within  a  conic. 

If  (x'f  y')  is  on  the  conic,  the  two  tangents  from  it  will  coincide, 
and  each  of  the  points  (iCi,  y^)  and  (x2,  y^  will  coincide  with  {x\  y'). 
Tlierefore  the  tangent  is  the  particular  case  of  the  polar  which  passes 
through  its  oimipole.     (See  demonstration  in  §  169.) 

95.   If  the  polar  of  a  point  P\x\  y')  pass  through  P"(x"j  y"),  then 
will  the  polar  of  P"  pass  through  P'.     (See  fig.  §  94.) 
Let  the  equation  of  the  conic  be  [§  93,  (1)] 

ax'  +  y'-^2gx-^g'  =  0.  (1) 

The  equations  of  the  polars  of  P'  and  P"  are 

axx'  +  yy'  +g(x  +  x')+g'  =  0  (2) 

and  axx"  +  yy"  +  g(x  +  x")  +  g'  =  0.     (§94.)  (3) 

The  line  (2)  will  pass  through  the  point  P"  if 

ax'x"-j-yY+g{x'  +  x")  +  g'  =  0;  (4) 

but  this  is  also  the  condition  that  (3)  shall  pass  through  P',  which 
proves  the  proposition. 


CoR.  I.     The  locus  of  the  poles  of  all  lines  passing  through  a  fixed 
point  is  a  straight  line;  viz.  the  polar  of  the  fixed  point. 

CoR.  II.     If  the  polars  of  two  points  P  and  Q  meet  in  i?,  then  R  is 
the  pole  of  the  line  PQ. 


130  CONIC   SECTIONS  [95 

Two  straight  lines  are  said  to  be  conjugate  with  respect  to  a  conic 
when  each  passes  through  the  pole  of  the  other. 

Two  points  are  said  to  be  conjugate  with  respect  to  a  conic  when 
each  lies  on  the  polar  of  the  other. 


EXAMPLES 

Find  the  equations  of  the  tangent  and  normal  to 

1.  x^  =  2y,  at  (-2,  2).  2.   y^  =  Sx,  at  (2,  -4). 

3.  x2  +  ?/2  =  25,  at  (4,  -  3).  4.  x^-y^  =  16,  at  (-  5,  3). 

5.  ic2  +  4  ?/2  =  8,  at  (-  2,  1).  6.2  y'^  ~-x'^  =  4,  at  (2,  -  2). 

Find  the  equations  of  the  tangents  to  the  following  conies  at  the  origin  : 

7.  x2  +  ?/2  +  2x  =  0.  S.  x^  +  2x  +  3y  =  0. 

9.  2xy  +  bx-3y  =  0.  10.  Sx^ -2xy -\-4:X-2y  =  0. 

11.  State  a  rule  for  finding  the  tangent  to  a  conic  at  the  origin, 
ind  the  polar  of  the  point 

12.  (3,  2)  with  respect  to  y^  =  6  x. 

13.  (  —  2,  —  4)  with  respect  to  x^  +  ?/2  =  4. 

14.  (1,  1)  with  respect  to  2x^ -\-Sy^  =  l. 

15.  (0,  0)  with  respect  to  2  x^  -  3  y2  _|_  12  x  -  6  y  +  21  =  0. 

16.  Give  a  rule  for  writing  the  equation  of  the  polar  of  the  origin. 

Find  the  tangents  to  the  following  conies  drawn  from  the  given  points  (see 


A      IS 


K 


'    17.  1/2  =  4 X,  (2,3).  18.  y^  =  6x,  (-3,  -1). 

19.   x2  +  ?/2  =  25,  (-1,7).  20.   9x2  +  25  2/2  =  225,  (10,  -3). 

^  21.   Show  that  the  polar  of  the  focus  is  the  directrix. 

What  is  the  locus  of  the  intersection  of  tangents  at  the  ends  of  focal  chords  ? 
(Use  equation  (1),  §  93.) 

22.  Show  that  the  line  joining  the  focus  to  any  point  on  the  directrix  is  per- 
pendicular to  the  polar  of  the  latter  point. 

23.  Show  that  tangents  to  a  conic  at  the  ends  of  a  chord  through  the  centre 
are  parallel. 

24.  What  is  the  polar  of  the  centre  of  a  conic  ?     Where  is  the  pole  of  a  line 
passing  through  the  centre  ? 

\i'     26.   What  is  the  pole  of  a;  cos  a  4- J/ since  =^  with  respect  to 

x^  +  y'^  =  r^?    y'^  =  2x? 


\i     2 


CHAPTER   VIII  

THE  PARABOLA 

96.   Standard  /equations  of  the  tangent,  polar,  and  normal  to  the 
parabola. 

In  studying  the  properties  of  the  parabola  in  this  chapter  we  shall 
use  the  standard  form  of  the  equation  found  in  §  88,  viz. 

2/2  =  4  ax.  (1) 

Then  the  focus  is  the  point  (a,  0),  the  directrix  is  the  line x=  —  a, 
and  the  latus  rectum  is  4  a. 

Equation  (6),  §  92,  applied  to  (1)  gives 

yy'  =2a(i€  +  ic'),  (2) 

as  the  equation  of  the  tangent  at  the  point  («',  y'),  if  (»',  y*)  is  on 
the  curve;  but  always  the  equation  of  the  polar  of  (x,  y'),  (§  94), 
with  respect  to  the  parabola  (1). 

The  equation  of  the  normal  at  the  point  (x',  y')  on  the  curve  is 
[(2),  §  62] 

II  _  1/'  =    — 

2a 

or  2  a{y  -  y')  4-  y\oc  -  a?')  =  0.  (4) 

The  tangent  at  the  vertex  (0,  0)  is  the  line  a;  =  0 ;  and  the  normal 
at  the  same  point  is  y  =  0,  i.e.  the  axis  of  the  curve. 
Ex.  1.     Show  that  the  equation  of  the  parabola  is 
2/2  =  4  a(x  ±  a), 
according  as  the  origin  is  at  the  focus  or  on  the  directrix. 
Ex.  2.    Change  the  equations  of  the  parabolas 

(y  -ky  =  i  a(x  -  h)  and  (x  -hy  =  4  a(y  -  k) 
to  the  standard  form,  and  show  that  their  vertices  are  at  the  point  (h,  k). 

Ex.  3.     What  relation  does  the  line  (3)  have  to  the  parabola  when  the  point 
(x'iy')  is  not  on  the  curve  ? 

181 


y-y'  =  -:^(^-^')^  (3) 


132 


THE   PARABOLA 


[97 


97.    Geometnc  properties  of  the  parabola. 


M 
R 

^                                              P-X^^^ 

\ 

\      X 

T                                   D 

O 

If                         n            g 

Let  the  tangent  at  the  point  P{x\  if)  meet  the  axis  in  jT,  the 
directrix  in  i2,  and  the  tangent  at  the  vertex  in  Q.     Let  PM  and 

PN  be  the  perpendiculars  from  P  to  the  directrix  and  axis,  re- 
spectively. 

Let  the  normal  at  P  meet  the  axis  in  G, 

Then  we  have  the  following  properties  ; 

TO=:ON=x\                                         [(2),  §96.]  (1) 

.-.     Subtangent  =  T]Sr=  2  0N=  2  x'.  (2) 

OQ  =  iNP=^y'.  (3) 

TF=FP=FG  =  a-hx'.  (4) 

Z.FPR  =  ZMPR.  (5) 

Z  RFP  =  Z  BMP  =  i  ,r.          (See  Ex.  22,  p.  130.)  (G) 

FM  is  perpendicular  to  TP.  (7) 

FM,  PT,  and  OY  meet  in  a  point  (8) 

0G  =  2a  +  x\                                         [(4),  §96.]  (9) 

.-.     Subnormal  =  NG  =  2  a,  a  constant.  (10) 


98]  THE   PARABOLA  133 

The  use  of  parabolic  reflectors  depends  on  the  property  expressed 
in  (5).     Let  the  student  explain. 

Properties  (5)  and  (7)  suggest  a  method  of  drawing  tangents  from 
an  exterior  point.     Show  how  this  can  be  done. 

98.  Equations  of  the  tangent  and  normal  in  terms  of  the  slope  m. 
The  equation  of  the  tangent  [(2),  §  96]  may  be  written 

2a     ,  2 ax'     2a     ,  4: ax'  ,^. 

Let  — p  =  m ;  then  ^  =  — ,  and  (2)  may  be  written 

l/  =  ^^^^j  (3) 

which  is  the  required  equation.     That  is,  the  line  (3)  will  touch  the 
parabola  y^  =  4  ax,  whatever  the  value  of  m  may  be. 

In  a  similar  manner  it  can  be  shown  from  (3),  §  96,  that  the  equa- 
tion of  the  normal  expressed  in  terms  of  its  slope  is 

y  =  mx  —  2  atn  —  atn^*  (4) 


EXAMPLES 

1.  Find  the  equations  of  the  tangents,  and  the  normals  at  the  ends  of  the 
latus  rectum. 

2.  Show  that  the  line  y  =  3  a;  4-  -  touches  the  parabola  y'^  =  Aax',  and  also  that 
y  =  Ax-\--  touches  y'^  =  S  ax. 

3.  Find  the  equation  of  the  tangent  to  y^  =  12  x  which  makes  an  angle  of  60°  ""^ 
with  the  X-axis. 

4.  Find  the  tangent  to  the  parabola  y^  =  6  a;  which  makes  an  angle  of  45°  with 
the  X-axis. 

Find  the  coordinates  of  the  vertex,  of  the  focus,  the  length  of  the  latus  rectum, 
and  the  equation  of  the  directrix  of  each  of  the  following  parabolas : 

6.   y2_3a;^.6.  6.   x2  +  4x  +  2y  =  0.  7.   (y- 4)2  =  6(x  + 2). 

8.   4(x-3)2  =  3(y-M).  .  9.   y^-l- 8x- 6y-M  =  0. 


\Xm 


134 


THE   PARABOLA 


[99 


99.    Tlie  locus  of  the  middle  points  of  a  system  of  parallel  chords  of  a 
parabola  is  a  straight  line  parallel  to  the  axis  of  the  parabola. 


Let  ABhe  any  one  of  the  chords,  let  P'(x'j  y')  be  its  middle  point, 
and  let  y  be  the  angle  it  makes  with  the  axis  of  the  parabola. 
Then  the  equation  of  AB  may  be  written  [(4),  §  43] 


x  —  x'     y  —  y 


=  r, 


(1) 
(2) 


cos  y         Sin  y 

or  a;  =  ic' -|- r  COS  y,     y  =  y' -{- r  sin  y. 

Let  the  equation  of  the  parabola  be 

y'  =  4.ax.  (3) 

Substituting  in  (3)  the  values  of  x  and  y  given  by  (2),  we  have  for 
the  points  common  to  the  chord  and  the  curve 

(y'  +  r  sin  y)^  =  4  a  (x'  -\-  r  cos  y), 
or  7^sm^y-^2(y'siny  —  2aGosy)r-{-y'^  —  4:ax'=:0j  (4) 

a  quadratic  equation  in  r,  whose  roots  are  represented  by  the  dis- 
tances P'B  and  P'A.  Since  P'  is  the  middle  point  of  AB,  the  sum 
of  these  roots  is  zero.     That  is, 

2/' sin y  — 2a cosy  =  0.  (§  ^8.) 

2a 


Whence 
where  m  is  the  constant  slope  of  the  chords. 


y'=2a  cot  7  =  —  J 
'       m 


(5) 


100]  THE   PARABOLA  135 

The  coordinates  of  P'  therefore  satisfy  the  equation 

y=^=2acoty.  (6) 

Hence  the  locus  of  P',  as  AB  moves  keeping  m  constant,  is  a 
straight  line  O'X'  parallel  to  the  axis  of  the  parabola. 

Definition.  The  locus  of  the  middle  points  of  a  system  of 
parallel  chords  of  a  conic  is  called  a  Diameter;  and  the  chords  it 
bisects  are  oblique  double  ordinates  to  that  diameter  considered  as 
an  axis  of  abscissas. 

We  have  seen  in  §  93  that  a  diameter  of  a  parabola  meets  the 
curve  in  only  one  point  at  a  finite  distance  from  the  directrix.  This 
point  is  called  the  Extremity  of  the  diameter. 

CoR.     The  line  (6)  meets  the  curve  in  0'  where 

x  =  —^  =  EO',    y  = (7) 

The  equation  of  the  tangent  at  0'  is,  therefore  [(2),  §  96], 

2/  =  ^^^  +  ^-  (8) 

Hence  the  tangent  at  the  extremity  of  a  diameter  is  parallel  to  the 
chords  bisected  by  that  diameter. 

100.  To  find  the  equation  of  a  parabola  ivhen  the  axes  are  any 
diameter  and  the  tangent  at  its  extremity. 

Using  the  figure  of  §  99,  and  keeping  the  same  notation,  we  will 
let  0'P'  =  x,  the  new  abscissa,  and  P'B  =  yf  the  new  ordinate. 

Then  y  is  always  the  same  as  r  of  equation  (4),  §  99.  And  since 
the  coefficient  of  the  first  power  of  r  in  this  equation  is  zero,  we  have 


where 
and 


„     4:ax'  —  y'^ 
^-     sin^y    ' 

(1) 

,     2a 
m 

[(5),  §  99.] 

f  =  RO'-\-0'P'  =  —, 

rn? 

+  x. 

[(7),  §  99.] 

„       4a 

.'.  y^=    .    „    X. 

^       sin^'y 

(2) 

56 

THE 

PARABOLA 

[100 

Now 

Fa-- 

1  +  tan^r 

a 

[(4),  §  97.] 

(3) 

-  Oi 7) — 

tan^y 

sin^y 

Therefore, 

if  a': 

sm^y 

',  the  required  equation  is 

2/2  =  4  a'x.  (4) 

Hence  the  equation  y^  —  4iax  always  represents  a  parabola,  the 
a>axis  being  a  diameter,  the  ^/-axis  the  tangent  at  its  extremity,  a  the 
distance  from  the  focus  to  the  origin,  and  4  a  the  length  of  the  focal 
chord  parallel  to  the  ^/-axis. 

Formula  (6),  §  92,  by  means  of  which  equation  (2),  §  96,  was 
obtained,  and  also  the  derivation  of  equation  (3),  §  98,  from  equation 
(2),  §  96,  hold  good  equally  whether  the  axes  are  rectangular  or  not. 
That  is,  if  the  equation  of  a  parabola  is  2/^  =  4  ax,  the  line 

yy^  =  2a{x-\-x^)  (5) 

will  be  the  tangent  at  the  point  {x\  y')  if  the  point  is  on  the  curve ; 

but  always  the  polar  of  (ic',  y')  with  respect  to  the  parabola.     And 

the  line  ^ 

y  =  mx-\-—  (6) 

will  also  touch  the  parabola  for  all  values  of  m,  the  meaning  of  d 
being  that  given  in  §  50. 

Cor.  The  polar  of  any  j^oint  with  respect  to  a  parabola  is  parallel  tr> 
the  chords  bisected  by  the  diameter  through  the  point. 

Conversely,  the  locus  of  the  poles  of  parallel  chords  is  the  bisecting 
diameter. 

For  the  polar  of  any  point  {x\  0)  is,  by  (5),  x  =  —  x\ 

EXAMPLES  ON  CHAPTER  VIII 

1.  Find  the  equation  of  that  chord  of  the  parabola  y'^  —  Qx  which  is  bisected 
by  the  point  (4,  3). 

/    2.   Find   the  equation  of   the  chord  of  x'^  =  —^y  whose   middle  point  is 
(-3,-2). 

3.  Find  the  equations  of  the  tangents  drawn  from  the  point  (—2,  2)  to  the 
parabola  y'^  =  Qx. 


100]  THE  PARABOLA  137 

4.  Show  that  the  axis  of  the  parabola  y^=:Sx  divides  each  of  the  chords 
whose  equations  are    .        p  =  — ottb  iijl^  two  segments  whose  product  is  64. 

6.  For  what  point  on  the  parabola  y^^'iaxm  (1)  the  subtangent  equal  to 
the  subnormal,  and  (2)  the  normal  equal  to  the  difference  between  the  sub- 
tangent  and  the  subnormal  ? 

'"'^  6.   Show  that  the  lines  y  =  jt  (a;  -f  2  a)  touch  both  the  parabola  y'^  —  %ax  and 
the  circle  a^  +,j/2  _  2  ^2.      '  '  ^ 

7^  Find  the  equation  of  the  common  tangent  to  the  parabolas  y"^  =  iiax  and 
»2  =  4  hy.  Show  also  that  if  a  =  &,  the  line  touches  both  at  the  end  of  the 
latus  rectum. 

E        8.   Two  equal  parabolas,  A  and  5,  have  the  same  vertex  and  their  axes  In 
opposite  directions.     Prove  that  the  locus  of  the  poles  with  respect  to  -B  of  tan- 
,    gents  to  A  is  the  parabola  A. 

9.  Show  that  the  locus  of  the  poles  of  tangents  to  the  parabola  y^  =  4:ax 
with  respect  to  the  parabola  y^=  4:bx  is  the  parabola  ay^  =  4  b^x. 

•\  y  10.   Show  that  for  all  values  of  m  the  line 
/  .     .     .   .  a 


/ 


1/^ 


y  —  m(x  +  a)  +  —  will  touch  y^z=:4  a(x  +  a); 

tTl 

y  =  m(x  —  a)  -h  —  will  touch  2/2  —  4  q^^x  —  a) ; 


and         (y  —  k)  =  m{x  —  h)  -\ —  will  touch  (y  —  k)'^  =  4  a(x  —  h"). 

11.   If  (a;',  y')  and  (ic",  y")  are  the  points  of  contact  of  two  tangents  to 
y2  =  4  ax,  show  that  the  coordinates  of  their  point  of  intersection  are 
X  =  Vx'x",  y  =  i(y'  +  y"). 

4'     12.   Show  that  the  directrix  is  the  locus  of  the  vertex  of  a  right  angle  whose 
sides  slide  upon  a  parabola.     (§  98. ) 

13.  Two  lines  are  perpendicular  to  one  another;  one  of  them  is  tangent  to 
?/2  =  4  a(x  +  a),  and  the  other  is  tangent  to  y^  =  4  6(a;  +  6)  ;  show  that  these 
lines  intersect  on  the  line  x  +  a  +  b  =  0. 

14.  Show  that  the  line  Ix  +  my  +  n  =  0  will  touch  the  parabola  2/^  =  4  ax, 
if  In  =  am^. 

15.  If  the  chord  PQR  passes  through  a  fixed  point  Q  on  the  axis  of  the 
parabola,  show  that  the  product  of  the  ordinates,  and  also  the  product  of 
the  abscissas  of  the  points  P  and  i?,  is  constant. 

T      16.   Find  the  coordinates  of  the  point  of  intersection  of  y  =  mx -\ and 

y  =  m'x  H — -. .     Show  that  the  locus  of  this  point  is  a  straight  line  if  mm   is 

m' 

constant.     What  is  the  locus  when  mm'  =  —  1  ? 


138  THE   PARABOLA  [100 

17.  K  perpendiculars  be  let  fall  on  any  tangent  to  a  parabola  from  two  points 
on  the  axis  which  are  equidistant  from  the  focus,  the  difference  of  their  squares 
will  be  constant. 

18.  The  vertex  ^  of  a  parabola  is  joined  to  any  point  P  on  the  curve,  and 
PQ  is  drawn  at  right  angles  to  AP  to  meet  the  axis  in  Q.  Prove  that  the 
projection  of  PQ  on  the  axis  is  always  equal  to  the  latus  rectum. 

19.  If  P,  Q,  and  B  be  three  points  on  a  parabola  whose  ordinates  are  in 
geometrical  progression,  the  tangents  at  P  and  B  will  meet  on  the  ordinate 

of  q. 

20.  Show  that  the  locus  of  the  intersection  of  two  tangents  to  a  parabola  at 
V  points  on  the  curve  whose  ordinates  are  in  a  constant  ratio  is  a  parabola. 

21.  Prove  that  the  circle  described  on  a  focal  radius  as  diameter  touches  the 
tangent  drawn  through  the  vertex. 

22.  Prove  that  the  circle  described  on  a  focal  chord  as  diameter  touches  the 
directrix. 

23.  Find  the  locus  of  the  point  of  intersection  of  two  tangents  to  a  parabola 
which  make  a  given  angle  a  with  one  another. 

If  a  =  45°,  show  that  the  locus  is  ?/2  —  4  ax  =  (x-\-  a)^. 

lia  =  60°,  show  that  the  locus  is  y-  -  3  x^  -  10  aa:  -  3  a^  =  0. 

[Suggestion.  The  line  y  =  mx  +  —  will  go  through  (x',  y')  if  m^'  —  my'  +  a  =  0. 
The  roots  of  this  equation  are  the  slopes  of  the  two  tangents  which  meet  in 
(x',  y').     Let  mi,  m^  be  these  roots,  then  see  §  68.] 

24.  The  two  tangents  from  a  point  P  to  the  parabola  y"^  =  4:  ax  make 
angles  tan-%i  and  tan-im2  with  the  ai-axis.  Find  the  locus  of  P,  (1)  when 
wij  +  mi  is  constant,  (2)  when  wii^  +  m^^  is  constant,  and  (3)  when  m\m2  is 
constant. 

25.  If  K  is  the  area  of  a  triangle  inscribed  in  the  parabola  y^  —  4  dx,  and 
K'  is  the  area  of  the  triangle  formed  by  the  tangents  at  the  vertices  of  the 
inscribed  triangle,  prove  that 

8  a^=r  16  aK'  =  (yi  ~  2/2)  (^2  ~  2/3)  (2/3  ~  Vi), 
where  2/1,  ?/2,  ys  are  the  ordinates  of  the  vertices  of  the  inscribed  triangle.     (See 
Ex.  11.) 

Find  the  locus  of  the  middle  points 

26.  Of  all  ordinates  of  a  parabola.  27.    Of  all  focal  radii. 

28.  Of  all  chords  through  the  fixed  point  (h,  k). 

As  special  cases,  let  (h,  k)  be  (1)  the  focus,  (2)  the  vertex,  (3)  the  point 
(4  a,  0),  and  (4)  the  point  (— a,  Q). 

29.  Show  that  the  parabola  is  concave  towards  its  axis. 


CHAPTER  IX  ^_ 

THE  CIRCLE 

XOl.  Equations  of  the  circle,  and  the  corresponding  equations  of  the 
tangentj  polar,  and  normal. 

We  have  seen  in  §  32  that  the  equation  of  the  circle  whose  radius 

is  r  takes  the  simple  form 

a^  +  /  =  v'2,  (1) 

when  the  origin  is  at  the  centre ;  while  if  the  centre  is  at  the  point 
(a,  b)  the  equation  may  be  written 

(x-ay-^(y-by='r'.  (2) 

Moreover,  we  have  found  in  §  87  that  the  locus  of  any  equation 
of  the  second  degree  is  a  conic.  Now  the  conic  represented  by  the 
general  equation  (5),  §  87,  will  be  a  circle  if  a  =  6  and  h  =  0.  For 
this  equation  may  then  be  written 

x^-\-y'  +  2gx  +  2fy-^c  =  0.  (3) 

Equation  (3)  may  be  put  in  the  form  of  (2),  which  gives 

{x  +  gy+(y+fy  =  9'-{-r-c.  (4) 

Hence  the  locus  of  (3)  is  a  circle  whose  centre  is  the  point 
(—  g,  — /),  and  the  radius  is  equal  to  ^ g^ -\-f^  —  c. 

The  circle  will  therefore  be  real,  a  point,  or  imaginary  according 
a'S(/2+/^-c>,  =,  or  <0. 

By  applying  the  rule  of  §  92  to  equations  (1),  (2),  and  (3),  re- 
pectively,  we  obtain 

xoc'  4-  yy'  =  r'^9  ® 

{X  -  a)(x'  -a)  +  {y-  h){y<  -b)=  r^,  (6) 

and  XX'  +  yy'  +  g(x  +  x')  +  f(y  +  y')  +  c  =  0.  (7) 

These  are  the  equations  of  the  tangent  to  the  circles  (1),  (2),  (3), 
respectively,  at  the  point  (x',  y')  if  this  point  is  on  the  curve;  but, 

139 


140  THE   CIRCLE  [101 

by  §  94,  they  are  always  the  equations  of  the  polar  of  the  point 
(x\  y')  with  respect  to  the  circles  represented  by  (1),  (2),  (3). 

Since  the  normal  (§  57)  at  any  point  (x',  y')  of  the  circle  a:F-\-y^  =  7^ 
is  perpendicular  to  (5),  its  equation  is  [(2),  §  62'] 

or  •  scy'  -  oc'y  =  0.  (8) 

That  is,  the  normal  at  any  point  of  a  circle  passes  through  the 
centre. 

The  equations  of  the  normals  to  the  circles  (2)  and  (3)  at  the 
point  (x'j  2/')  are,  respectively  [(2),  §  62], 

y-y'  =  ^\^(x-x'),  (9) 

x'  —  a 

and  y-y'  =  yL±f(x-x^):  (10) 

x'  +  g 

or  xy' —  x'y  —  b(x  —  x')-\-a(y  —  y')=0,  (11) 

and  xy'  —  x'y  -\-f(x  —  x')  —  g(y  —  y')  =  0.  (12) 

The  general  equation  of  the  circle  (3),  or  (2),  contains  three 
parameters,  or  constants.  Therefore  a  circle  can  be  made  to  satisfy 
three  conditions,  and  no  more.  If  we  wish  to  find  the  equation 
of  a  circle  which  satisfies  three  given  conditions,  we  assume  the 
equation  to  be  of  the  form  (3),  or  (2),  and  then  determine  the 
values  of  the  constants  g,  f,  c,  or  a,  b,  r,  from  the  given  conditions. 

Ex.    Find  the  equation  of  the  circle  passing  through  the  three  points  (0,  1), 

(2,0),  and  (0, -3). 

Let  the  equation  of  the  required  circle  be 

x^  +  y^  +  2gx  +  2fy  +  c  =  0.  (1) 

Since  the  given  points  are  on  the  circle,  their  coordinates  must  satisfy 
equation  (1). 

.-.    l  +  2/+c  =  0,     4  +  4^  +  c  =  0,     9-6/+c  =  0. 

Whence  we  find  gr  =  —  ^,  /=  1,  and  c  =  —  3.  Substituting  these  values  in 
(1)  the  required  equation  becomes  x^  +  y^  —  I  x  +2^-3=0. 

The  centre  is  the  point  (^,  -  1),  and  the  radius  is  ^VOd. 


102] 


THE   CIRCLE 


141 


102.  A  geometrical  construction  for  the  polar  of  a  point  with  respect 
to  a  circle. 


Let  the  equation  of  the  circle  be 

x'^,/  =  'i^,  (1) 

Let  P(x',  y')  be  any  point,  BC  its  polar,  and  let  OP  and  BC 
intersect  in  Q.     Then  the  equation  of  BC  is  [(5),  §  101] 

xx'  +  yy'  =  r^, 

and  the  equation  of  the  line  OP  is  (§  44) 

xy'  —  x'y  =  0. 

Hence  BC  is  perpendicular  to  OP  (§  45),  and  therefore 


(2) 
(3) 


0Q  = 


Vx^'  +  y'' 


Also 


[(5),  §47.] 

[(4),  §7.] 


(4) 


(5) 
(6) 


OP=Vx'^-{-y''. 

.-.  OP'  OQ  =  r'. 

We  therefore  have  the  following  construction  for  the  polar  of 
a  point  P.  Draw  OP  and  let  it  cut  the  circle  in  R;  then  con- 
struct a  third  proportional,  OQ,  to  OP  and  r,  i.e.  take  Q  on  the 
line  OP,  such  that  OP:  OP  =  OR:OQ,  and  draw  a  line  through  Q 
perpendicular  to  OP. 

Ex.  1.  Construct  the  pole  of  a  given  line. 


142  THE   CIRCLE  [103 

103.    To  find  the  equation  of  the  tangent  to  the  circle 

a?  +  f  =  i^  (1) 

in  terms  of  its  slope  m. 

The  line  y  =  mx  +  h  (2) 

will  touch  the  circle  (1)  if  the  perpendicular  distance  from  it  to  the 
origin  is  equal  to  the  radius  r  of  the  circle ;  that  is,  (§  47)  if 

^       :,  or  6  =  rVrT^.  (3) 


Vl  +  m^ 
Therefore  the  straight  line 

y  =  mx  +  ^Vl  +  in^  (4) 

will  touch  the  circle  (1)  for  all  values  of  m. 


Since  either  sign  may  be  given  to  the  radical  Vl  +  m^  in  (3),  it 
follows  that  there  are  two  tangents  to  the  circle  for  every  value  of  m ; 
i.e.  there  are  two  tangents  parallel  to  any  given  straight  line. 

Ex.  1.  Derive  equation  (3)  by  treating  (1)  and  (2)  simultaneously  and  taking 
the  condition  for  equal  roots. 

EXAMPLES 

Find  the  equation  of  the  circle  passing  through  the  three  points 

1.    (1,  0),  (6,  0),  (0,  4).  2.    (0,  0),   (1,  1),  (4,  0). 

3.    (2,  -3),  (3,  -4),   (-2,  -1).  4.    (1,2),   (3,  -4),  (5,6). 

Find  the  equations  of  the  tangents  to  the  circle 

5.   x2  +  2/2  =  4  parallel  to2x  +  3?/  +  l=0. 
1^6.   a;2  +  ^2  _  6  a;  parallel  to3x-2y  +  2  =  0. 

Find  the  polar  of  the  point 

7.  (1,  2)  with  respect  to  a;2  +  y2  _  5. 

8.  (3,  -  2)  with  respect  to  3(x2  +  y2)  _  14. 

9.  (-4,  1)  with  respect  to  ic2  +  y2  _  2  aj  4-  6 y  +  7  =  0. 
Find  the  polar  of  the  line 

10.  2 oj  +  y  =  1  and  x  —  ^y  =  \  with  respect  to  ic2  +  ^2  _  2. 

11.  X  —  2  ?/  =  3  and  2  x  +  2/  =  4  with  respect  to  x^  +  if  =  6. 

12.  x  +  2/+l=0  with  respect  tox2  +  ?/2  +  4x-6i/  +  ll=0. 


\l.jr^ 


104] 


THE   CIRCLE 


143 


104.    To  find  the  length  of  a  tangent  drawn  from  a  given  point 
P{x\  y')  to  a  given  circle. 

Let  the  equation  of  the  circle  be 

(x-ay-h(y-by-r^  =  0.  ^    (1) 

Let  C  be  the  centre  and  PT  one  tangent  from  P. 


Then,  since  CPT  is  a  right  triangle, 
PT^=CP^-CT\ 
But  OT2=r2,  and  CP'=  (x'  -  ay  +  (y'  -  by.  [§  7,  (2).] 

.-.  PT^  =  («'  -  a)2  +  (2/'  -  6)2—^2.  (3) 

That  is,  the  square  of  the  tangent  is  found  by  substituting  the 
coordinates  x',  y'  of  the  given  point  in  the  left  member  of  equation  (1). 
Since  the  general  equation  of  the  circle, 

a^H-/  +  2i/x  +  2/2/  +  c  =  0,  (4) 

can  be  put  in  the  form  of  (1)  by  merely  adding  and  subtracting 
g^  and  /^  in  the  first  member,  it  follows  that  if  the  coordinates  of 
any  point  are  substituted  in  the  first  member  of  (4)  the  result  will  be 
equal  to  the  square  of  the  length  of  the  tangent  drawn  from  the  point 
to  the  circle ;  or  the  product  of  the  segments  of  any  chord  (or  secant)  • 

drawn  through  the  point.     (See  proof  of  §  154.)  >-       .-^ 


Ex.  1.   What  is  the  meaning  of  (3)  when  the  second  member  is  negative 

Ex.  2.   What  is  represented  by  c  in  equation  (4)? 

Ex.  3.  Where  is  the  origin  if  c  is  positive  ?  if  c  is  zero  ?  if  c  is  negative  ? 


,^ 


144 


THE  CIRCLE 


[105 


105.  If  a  circle  passes  through  the  common  points  of  two  given  circles, 
tangents  drawn  from  a7iy  point  on  it  to  the  two  given  circles  are  in  a 
constant  ratio. 


Let  S  =  x^-^y'-h2gx  +  2fy-\-c  =  0  (1) 

and  S'^x'-\-y^-\-2g'x-\-2f'y-\-c'  =  0,  (2) 

be  the  equations  of  the  two  given  circles. 

Then  the  locus  of  aS'  =  kS^,  i.e.  (See  Ex.  5,  p.  62.) 

a?-\-f-^2gx  +  2fy-\-c  =  X(x'  +  f  +  2g'x-\-2fy-\-c'),         (8) 

for  all  values  of  X,  will  pass  through  the  common  points  A,  B,  of 
(1)  and  (2).  Moreover,  (3)  is  a  circle  (§  101),  and  therefore,  for 
different  values  of  X,  represents  all  circles  through  the  intersection 
of  (1)  and  (2). 

Let  P(x',  y')  be  any  point  on  (3) ;  let  PT  and  PT^  be  the  tangents 
to  (1)  and  (2)  respectively.  Then  the  coordinates  x',  y'  must  satisfy 
(3),  and  we  therefore  have 

x"  -\-y"  +  2  gx'  +  2/?/'  +  c  =  \(x''  +  y"-^2  g'x'  +  2f'y'  +  c').     (4) 

Therefore  PT'  =  X'PT'%  (§104.)      (5) 

which  proves  the  proposition,  since  X  is  constant  for  any  particular 
circle. 


106]  THE  CIRCLE  145 

When  A.  =  1,  it  is  easy  to  show  that  the  radius  and  the  coordinates 
of  the  centre  (§  101)  of  the  circle  represented  by  equation  (3)  all 
become  infinite.     In  this  case  the  equation  reduces  to 

2(flr-fir')«  +  2(/-/')2/  +  c-c'  =  0,  _^  (6) 

which  is  of  the  first  degree,  and  therefore  represents  the  straight 
line  AB  through  the  common  points  of  the  two  given  circles. 

Let  QR  and  QR'  be  tangents  to  /iS  =  0  and  S'  =  0,  respectively, 
from  any  point  Q  on  AB]  then,  since  ABQ  is  the  circle  through  the 
common  points  of  (1)  and  (2)  corresponding  to  X  =  l,  it  follows  from 
(5)  that 

QR  =  QR'.  (7) 

That  is,  tangents  drawn  to  the  two  given  circles  from  any  point 
on  the  line  (6)  are  equal. 

It  is  to  be  noticed  that  the  straight  line  given  by  (6)  is  in  all  cases 
real,  provided  g,  f,  c,  g\  f\  c'  are  real,  although  the  circles  /S'  =  0  and 
>S'  =  0  may  not  intersect  in  real  points ;  in  fact  one  or  both  of  the 
circles  may  be  wholly  imaginary.  We  have  here,  therefore,  the  case 
of  a  real  stiaijght  line  passing  through  the  imaginary  points  of  inter- 
isection  of  two  real  or  imaginary  circles.     (Cy.  §  94.) 

Definition.  The  straight  line  through  the  points  of  intersection 
(real  or  imaginary)  of  two  circles  is  called  the  Radical  Axis  of  the 
tAvo  circles. 

From  equation  (7)  it  follows  that  the  radical  axis  may  also  be 
defined  as  the  locus  of  the  points  from  which  tangents  drawn  to  the 
two  circles  are  equal  to  one  another. 

Cor.  If  the  coefficients  of  o^  in  S  and  S'  are  unity,  the  equation  of 
the  radical  axis  of  the  two  circles  S  =  0  and  S'  =  0  is  S  —  S'  =  0. 

Ex.  1.  Show  that  the  radical  axis  of  two  circles  is  perpendicular  to  the  line 
joining  their  centres. 

Ex.  2.  If  tangents  are  drawn  to  two  circles  from  any  point  on  a  line  parallel 
to  their  radical  axis,  show  that  the  difference  of  the  squares  of  these  tangents 
is  constant. 

Ex.  3.  Show  that  the  radical  axis  of  two  circles  divides  the  line  joining  their 
centres  into  two  segments,  such  that  the  difference  of  their  squares  is  equal  to 
the  difference  of  the  squares  of  the  radii. 


146  THE   CIRCLE  [106 

106.    The  radical  axes  of  three  circles,  taken  in  pairs,  meet  in  a  point. 

Let  Si  =  0,  /iS'2  =  0,  ^iSg  =  0  be  the  equations  of  three  circles,  in  each 
of  which  the  coefficient  of  ic^  is  unity. 

Then  the  equations  of  their  three  radical  axes  are  (§  105,  Cor.) 

Si  —  /S'2  =  0,     S2  —  ^3  =  0,     Sq  —  Si  =  0. 

The  sum  of  any  two  of  these  equations  is  equivalent  to  the  third. 
Hence  they  form  a  consistent  system,  and  therefore  their  loci  meet 
in  a  point.     Or,  prove  by  §  49,  letting  A  =  1. 

TMs  point  is  called  the  Radical  Centre  of  the  three  circles. 

EXAMPLES  ON   CHAPTER  IX 

Find  the  length  of  the  tangents  (or  the  product  of  the  segments  of  the  chords^ 
drawn  from  the  points 

V    1.   (3,  2),  (5,  -  4)  to  the  circle  x"^  +  y^  =  4. 

2.  (-  3,  2),  (4,  -  4)  to  the  circle  x^ -h  y^  =  25. 

3.  (3,  -  2),  (1,  3)  to  the  circle  x^ -\- y"^  -  2x  -  4y  =  0. 

i   4.  (2,  1),  (0,  0)  to  the  circle  2  (x2  +  ?/)  _  12  x  -  4  y  +  15  =  0. 

6.  (0,  0),  (-  2,  -  5)  to  the  circle  x'^ -j- y^  -  6x -\- 4y  +  4  z=0. 

6.  (0,  0),  (6,  -  3)  to  the  circle  x'^  +  y"^  -\- 6  x  -  8y  -  U  =0. 

Find  the  radical  axis  of  the  circles 

I   7.  x2  +  ?/2  +  6  x  -  4  ?/  -  3  =  0  and  a:2  +  ?/2  -  4  X  +  8  ?/  -  5  =  0. 

8.  a;2  +  ?/2  -  8  X  -  10  y  +  25  =  0  and  x2  +?/2+8x-2y  +  8  =  0. 

9.  x2  +  ?/2  +  ax  +  6y  -  c  =  0  and  rtx2  +  ay"^  +  a^x  +  b^y  =  0. 

10.  Find  the  radical  axis  and  the,  length  of  the  common  chord  of  the  circles 

x^  +  y^  +  ax-\-by  +  c  =  o'and  x^  +  y^ -^  bx  +  ay  +  c  =  0. 

11.  Show  that  the  three  circles 

x2  +  y2_2x-4y  =  0,  x2  +  2/2_6x  +  4^  +  4  =  0, 

x2  +  ?/2_8x  +  8i/  +  6  =  0 
have  a  common  radical  axis.     Find  the  equation  of  a  fourth  circle  such  that  the 
four  shall  have  a  common  radical  axis. 

Find  the  radical  centre  of  the  three  circles 

12.  af2  +  2/2  _  4  a;  ^  8  y  _  5  ^  0,  x2  +  y2  _  g  x  -  10  ?/  +  25  =  0, 
'                                         x2  +  y2  +  8x  +  ll?/- 10  =  0. 

13.  x2-}-y2_|.6x-8?/  +  9  =  0,  x2+?/2  +  8x  +  2y  +  9  =  0, 

2(x2+2/2)_5(3a;  +  y)  +  18  =  0. 


hMA 


106]  THE  CIRCLE  147 

/ 

14.  What  is  the  equation  of  the  normal  in  terms  of  its  slope  ? 

^  16.  How  many  normals  can  be  drawn  from  a  point  to  a  circle  ? 

16.  Find  the  equation  of  a  circle  passing  through  (0,  4)  and  (6,  0),  and  hav- 
ing V 13  for  radius. 

9    17.  Find  the  equation  of  a  circle  whose  centre  is  (3,  4)  and  which  tmiches 
the  line  4x-3y  +  20  =  0. 

18.  Find  the  equation  of  the  circle  passing  through  the  point  ( —  3,  6)  and 
touching  both  axes. 
^  19.  Find  the  equation  of  the  circle  touching  the  line  y  —  c  and  both  axes. 

Write  down  the  equation  of  the  tangent  to  the  circle 

J80.  x2  +  y2  _  2  a;  +  3  y  -  4  =  0  at  the  point  (2,  1). 
'^21.  a;2  ^_  y2  ^_  4  aj  _  6  y  _  13  =  0  at  the  point  (-  3,  -  2). 

22.  Show  that  the  lines  y  =  m{x  —  r)  ±r  V 1  +  wi^  touch  the  circle 

x2  +  ?/2  =  2  rx, 

whatever  the  value  of  m  may  be. 

Find  the  equation  of  the  tangent  to  the  circle 

23.  9  (x2  +  y2)  _  9  (6  X  -  8  y)  +  125  =  0  parallel  to  3  x  +  4  j^  =  0. 

24.  Show  that  the  line  x  —  2  y  =  0  touches  the  circle 

a;2  _|_  ^2  _  4  a;  4-  8  2/  =  0. 
-4.    25.  The  line  y  =  3  a;  —  9  touches  the  circle 

ic2_|_2/2  +  2x  +  4y-6  =  0. 
Find  the  coordinates  of  the  point  of  contact. 

26.  Find  the  equation  of  the  tangent  to  x2  -f  2/2  _  ,.2  ^j)  which  is  perpendic- 
ular to  y  =  mx  4-  6,  (2)  which  passes  through  the  point  (c,  0),  (3)  which  makes 
with  the  axes  a  triangle  whose  area  is  ifi. 

Find  the  polar  of  the  point 
.      27.   (2,  -  \)  with  respect  to  x2  +  2/2  +  3  x  -  5  ?/  +  3  =  0. 

28.   (-  a,  6)  with  respect  to  x2  +  ?/2  -  2  ax  +  2  6y  +  a2  -  52  =  0. 
Find  the  pole  of  the  line 
•(  ^      29.  2  X  -I- 14  2/  =  15  with  respect  to  2  (x2  +  ?/2)  _  3  ^j  4.  5  j,  _  2  =  0. 

30.  3  (ax  -  6y)  -a^^-lfi  with  respect  to  x2  +  2/2  _  2  ax  -f-  2  6?/  =  a2  -l-  h\ 
^     31.  Show  that  the  circles  x2  +  ?/2_4a;^2i/  =  15  and  x^  ■\-y'^  —  ^  touch  one 
another  at  the  point  (—2,  1). 

32.  Show  that  the  radical  axis  of  two  circles  bisects  their  four  common  tan- 
gents. 

33.  The  distances  of  two  points  from  the  centre  of  a  circle  are  proportional 
to  the  distances  of  each  from  the  polar  of  *      other. 


148  THE  CIRCLE  [106 


/ 


"^    34.  What  is  the  analytic  condition  that  the  origin  shall  be  the  radical  centre 
of  three  given  circles  ? 

v^       35.   Find  the  equation  of  the  circle  through  the  origin  and  the  points  of  inter- 
section of  the  circles 

x'2-\-y'i-^x-1y  +  6  =  0  and  x^  +  y"^  +  ix  +  dy  -  12  =  0. 

-;   What  is  the  ratio  of  the  tangents  drawn  from  any  point  on  it  to  the  two  given 
circles  ? 
/  >^  36.  Find  the  equation  of  the  circle  which  touches  the  line  4  ?/  =  3  x  and  passes 
through  the  common  points  of 

x2  4-2/2  =  9  and  ic^  +  ?/2;+  x  +  2y=U. 

37.  What  is  the  ratio  of  the  tangents  drawn  from  any  point  on  the  third 
circle  in  Ex.  11  to  the  other  two  circles  ? 

^_    38.   Find  the  equations  of  the  straight  lines  which  touch  both  of  the  circles 
x2  +  2/2  =  4  and  (x  -  4)2  -f  ^2  _  i,  ^^g^  Sx±V7  y  =  S  and  x  ±  vT5  y  =  8. 

39.  Find  the  equations  of  the  common  tangents  to  the  circles 

x'^  +  y^-\-6y-\-  6  =  0  and  x"^ -h  y^  -  12  y  -\- 20  =  0. 

40.  If  the  length  of  the  tangent  from  the  point  (x',  y')  to  the  circle  x2  +  ?/2  =  9 
is  twice  the  length  of  the  tangent  from  the  same  point  to  x2  +  2/2  _j_  3  ^j  _  g  ^  _  q^ 
show  that  ^,2  +  2/'2  +  4  X'  -  8  y'  +  3  =  0. 

41.  If  the  tangent  from  P  to  the  circle  x2  -i-  ?/2  -f  3  y  =  0  is  four  times  as  long 
as  the  tangent  from  P  to  the  circle  x^  +  y^  =  9,  show  that  the  locus  of  P  is 

5(x2-f?/2)  =  ?/  +  48. 

42.  The  length  of  a  tangent  drawn  from  a  point  P  to  the  circle 

x'^  +  y'^+4:X-6y  +  4  =  0 

is  three  times  the  length  of  the  tangent  from  P  to  the  circle 

x2  +  2/2-6x  +  2?/  +  6  =  0. 
Find  the  locus  of  P. 

43.  Find  the  locus  of  a  point  whose  distance  from  the  origin  is  equal  to  the 
length  of  the  tangent  drawn  from  it  to  the  circle 

x2  +  ^2_8x-4?/  +  4  =  0. 

44.  Find  the  locus  of  a  point  P  whose  distance  from  a  fixed  point  is  in  a 
constant  ratio  to  the  tangent  drawn  from  P  to  a  given  circle.  Under  what 
condition  is  the  locus  a  straight  line  ? 

46.  Show  that  the  polar  of  any  point  on  the  circle 

x2  +  2,2  _  2  ax  -  3  a2  =  0, 

with  respect  to  the  circle  x^  +  ^2  ^  2  ax  -  3  a^  =  0,  will  touch  the  parabola 
2/2  +  4  ax  =  0. 


106]  THE  CIRCLE  149 

46.  Show  that  the  polars  of  the  point  (1,  0)  with  respect  to  the  two  circles 
x^  -^  y^  +  4:X  —  14:  =  0  and  x^  -\-  y^  =  4  are  the  same  line  ;  show  that  the  same  is 
true  of  the  point  (4,  0). 

47.  Find  two  points  such  that  the  polars  of  each  with  respect  to  the  two 
circles  x'^  +  y^-2x-S  =  0  and  x^  -{- y^ -i- 2  x  -  17  =  0  coincide. 

48.  A  certain  point  has  the  same  polar  with  respect  to  two  circles ;  prove  that 
any  common  tangent  subtends  a  right  angle  at  that  point.  Show  also  that  there 
are  two  such  points  for  any  two  circles. 

49.  Find  the  locus  of  the  intersection  of  two  tangents  to  jc2  4.  ^2  _  ,.2  which 
are  at  right  angles  to  one  another. 

50.  Find  the  locus  of  the  intersection  of  two  tangents  to  x^  -\-  y^  =  r^  which 
intersect  at  an  angle  a. 

61.  Show  that  if  the  coordinates  of  the  extremities  of  a  diameter  of  a  circle 
are  (xi,  yi)  and  (x2,  y^),  respectively,  the  equation  of  the  circle  will  be 

{x  -  xi){x  -  X2)  -\-  (y  ~  y\) (y  -  2/2)  =  0. 
[Suggestion.     Lines  joining  any  point  (a;,  y)  on  the  circle  to  (xi,  yi)  and 
(X'z,  y-z)  are  at  right  angles  to  one  another.] 

Find  the  equation  of  the  circle  which  touches 

62.  the  lines  x  =  0,  x  =  a,  and  3y  =  4x  +  3a. 

OneAns.    4  {x^  +  y^)- 4:  a  {x-h  6  y)+ 26  a^  =  0. 

53.   both  axes  and  the  line  -  +  |  =  1. 
a     0 

64.   Prove  analytically  that  the  locus  of  the  middle  points  of  a  system  of 

parallel  chords  of  a  circle  is  the  diameter  perpendicular  to  the  chords.     (See 

§  99.) 

'    65.   Show  that  as  a  varies  the  locus  of  the  intersection  of  the  lines 

X  cos  a  +  y  sin  cc  =  a  and  a;  sin  a  —  y  cos  a  =  b 
is  a  circle. 

66.  A  circle  touches  the  y-axis  and  cuts  off  a  constant  length  (2  a)  from  the 
X-axis  ;  show  that  the  locus  of  its  centre  is  x'^  —  y^  =  a^. 

57.  Two  lines  are  drawn  through  the  points  (a,  0)  and  (  —  ,a,  0)  and  make  an 
angle  a  with  one  another.     Show  that  the  locus  of  their  point  of  intersection  is 

x^  -\-y^  ±2  ay  cot  a  =  a^. 

58.  If  the  polar  of  the  point  (x',  y')  with  respect  to  the  circle  x^  +  y^  =  ^2 
touches  the  circle  x^-i-  y^  =  2  ax,  show  that  y'^  +  2  ax'  =  a^. 

59.  Show  that  if  the  axes  are  inclined  at  an  angle  w,  the  equation  of  the  circle 
is  (§  8)  (a;  _  a)2  +  (y  -  by  +  2  (x  -  a)  (y-  6)  cos  w  =  »^, 

where  (a,  6)  is  the  centre  and  r  the  radius. 


CHAPTER  X 
THE  ELLIPSE  AND  HYPERBOLA 

107.  Standard  equations  of  the  tangent^  polar,  and  normal  to  the 
ellipse  and  hyperbola. 

It  has  been  shown  in  §  89  and  §  90  *  that,  if  the  axes  of  the  curve 
are  taken  as  coordinate  axes,  the  equations  of  the  ceyitral  conies  may 
be  written  in  the  standard  form 

Then  the  coordinates  of  the  foci  are  (±  ae,  0);  the  equations  of 

a  2b^ 

the  directrices  are  «  =  ±  - ;  the  length  of  the  latus  rectum  is  —  ;  and 

e  = 

a 

For  equation  (1)  formula  (6),  §  92,  gives 

Equation  (2)  is  the  equation  of  the  polar  (§  94)  of  the  point  («',  y') 
with  respect  to  the  central  conic  (1),  which  polar  is  a  tangent  at  the 
point  {x'y  y')  when  (x',  y')  is  on  the  conic. 

The  equation  of  the  normal  at  any  point  (x',  y')  on  the  conic  (1)  is 

"-"'  =  ffc("-^'>'  "'  ^  =  "'•    f  (')'  «  ^2-]  (3) 

Ex.  1.  Find  the  equations  of  the  central  conies  when  the  origin  is  at  either 
focus  ;  at  either  vertex  ;  at  the  point  (^,  k) ,  the  coordinate  axes  being  parallel 
to  the  axes  of  the  conic. 

Ex.  2.  What  relation  does  the  line  (3)  have  to  the  conic  when  (x',  y')  is  not 
on  the  curve  ? 

*  These  sections  should  now  be  carefully  reviewed. 

t  We  shall  use  this  form  of  the  equation,  although  the  simpler  form  ax^  +  by^  =  1  is 
sometimes  more  convenient.  When  the  double  sign  db  or  =p  is  prefixed  to  b^,  the 
upper  sign  holds  for  the  ellipse  and  the  lower  for  the  hyperbola.  All  results  are  true 
for  both  curves  unless  the  contrary  is  expressly  stated.  Furthermore,  results  for  the 
ellipse  include  those  for  the  circle  as  the  special  case  when  a  =  6. 

160 


108]  THE  ELLIPSE   AND  HYPERBOLA  151 

108.    To  find  the  equation  of  the  tangent  to  the  conic 

in  terrns  of  its  slope  m. " 

Assume  the  equation  of  the  tangent  to  be 

y  =  mx  +  c,  (2) 

where  m  is  known,  and  c  is  to  he  determined  so  that  (1)  and  (2)  shall 
intersect  in  two  coincident  points  (§57).  t 

Eliminating  y  between  (1)  and  (2)  gives 

^  +  (^^  +  cf  _  -I  / 

a'  6^ 

or  x"  {a^m"  ±  6' )  +  2  ahmx  +  a^  {<?  T  6')  =  0.  (3) 

The  roots  of  equation  (3)  will  be  equal  if 

a2  (c2  IF  62)  (a  V  ^  j2>^  ^  ^a^^2 

Whence  c"  =  a'm^  ±  h\  (4) 

That  is,  the  points  of  intersection  of  the  straight  line  and  the 
conic  will  coincide  if 

c  =  ±  V  a'm'  ±  h\  (6) 

Hence  the  line  whose  equation  is 

y  =  mx  ±  V  a2m2  ±  62,  (6) 

will  touch  the  conic  (1)  for  all  values  of  m. 

The  double  sign  before  the  radical  in  (6)  shows  that  there  are  two 
tangents  for  every  value  of  m ;  i.e.  there  are  two  tangents  to  a  central 
conic  parallel  to  any  given  straight  line;  and  these  two  parallel  - 
tangents  are  equidistant  from  the  centre  of  the  conic. 

Ex.  1.     Derive  equation  (6)  by  the  method  used  in  §  98. 

Ex.  2.     In  a  similar  manner  show  that  the  equation  of  the  normal  to  (1) 

expressed  in  terms  of  its  slope  is 

w(a2  T  62) 

Va2±62w2 

Ex,  3.    How  many  normals  can  be  drawn  from  a  given  point  to  a  central 
conic  ? 


152 


THE  ELLIPSE  AND   HYPERBOLA 


[109 


109.    Geotnetric  properties  of  the  ellipse  and  hyperbola. 
Y  ^ 


Let  the  tangent  at  P(x',  y')  meet  the  axes  in  T  and  T";  let  the 
normal  at  P  meet  the  axes  in  N  and  N' ;  let  BP  be  the  ordinate  of 
P  and  F,  F'  the  foci  of  the  conic. 

Draw  FG,  F'G',  and  O/iT  perpendicular  to  the  tangent  PT. 

^^ThenOr=^,^  OT'^-^.^;.^   ^[(2),  §  107.] 


y 


a^—  X 


2/' 


.1^ 


0N=  eV,         ON'  =  ^^y'-  'l^^  [(3),  §  107.] 


Subnormal  =  i2iV=  (e^  —  l)x'  =  ?/ 


die' 


N^  T 


OK'NP=FG  .  JF^'G^'  =  ±  61 

PN^PM'=  FP.'F'P:=±  (a'  -  eV).      (§§  89,  90.) 

F^G  and  FG^  bisect  ^JV. 

The  locus  of  G^  and  G^'  is  a^  +  2/'  =  «^  [Use  (6),  §  108.] 


(1) 
(2) 
(3) 

(4) 

(P) 
(6) 
(7) 
(B) 
(^) 


F^N  __  F'  6  +777r2  ae  +  e^x'  ^  a  +  ex' 

]^F  "  OF-  0N~  ae-  e^x'  ~ a  -  ex' 

F'N^    F'P   y 

***      NF      ±FW 


(§§89,90.)     (10) 


109] 


THE   ELLIPSE   AND   HYPERBOLA 


153 


Therefore  the  tangent  and  the  normal  bisect  the  angles  between 
the  focal  radii  FP  and  F'P. 


Hence,  if  an  ellipse  and  a  hyperbola  have  the  same  foci,  the 
tangent  and  the  normal  to  one  of  the  curves  at  any  one  of  their  four 
common  points  are,  respectively,  the  normal  and  the  tangent  to  the 
other.     That  is,  the  two  conies  intersect  orthogonally. 

Conies  having  the  same  foci  are  called  Confocal  Conies. 

Ex.  1.  Explain  what  would  happen  if  a  light  were  placed' at  one  focus  of  an 
ellipse ;  a  hyperbola. 

Ex.  2.    What  is  the  limit  of  ON,  ON',  and  BN asx'  =  a?    asx'  =  0? 

Ex.  3.  Show  that  equations  (1),  (3),  and  OK'  NP=b^  are  also  true  when 
P  is  any  point,  TT'  the  polar  of  P,  and  PN  is  perpendicular  to  TT'. 


Ex.  4.    Show  that  the  equation 


1  represents  a  system  of  coit- 


al ^\     62  4.  X 

focal  conies,  where  X  is  the  arbitrary  parameter.  Investiijate  the  nature  of 
these  conies  for  values  of  X  ranging  from  —  oo  to  +  oo  .  Show  that  two  confo- 
cals,  an  ellipse  and  a  hyperbola,  pass  through  every  point  in  the  plane,  and  that 
these  meet  at  right  angles.  \ 


154  THE  ELLIPSE  AND  HYPERBOLA  [109 

EXAMPLES 
Find  the  eccentricity,  foci,  and  latus  rectum  of  each  of  the  following  conies  : 
1.  a;2  +  2  2/2=:4.  2.   ix^-9y^  =  S6. 

3.  4  x2  +  y2  =  8.  y   4.   3  a;2  -  2/2  =  9. 

f       6.    3(x- 1)2  +  4(^  +  2)2=1.  y/     6.    3(2/ -  1)2 -4(«+ 1)2  =  1. 

Find  the  equation  of  an  ellipse  referred  to  its  axes 

7.  if  the  latus  rectum  is  6  and  the  eccentricity  ^. 

8.  if  the  latus  rectum  is  4  and  the  minor  axis  is  equal  to  the  distance 
between  the  foci. 

9.  Find  the  equation  of  the  hyperbola  whose  foci  are  the  points  (+4,  0) 
and  whose  eccentricity  is  ■y/2. 

10.  Find  the  eccentricity  and  the  equation  of  the  ellipse,  if  the  latus  rectum 
is  equal  to  half  the  minor  axis. 

11.  Find  the  equation  of  the  hyperbola  with  eccentricity  2  which  passes 
through  (-4,  6). 

/     12.    Find  the  equation  of  the  ellipse  passing  through  the  points  (  —  2,  2)  and 
(3,  —  1);  also  the  equation  of  the  hyperbola  through  (1,  —  3)  and  (2,  4). 

Through  how  many  points  can  a  central  conic  be  made  to  pass  if  its  axes  are 
given  ?    Why  ? 

13.  Find  the  eccentricity  and  the  equation  of  a  central  conic  if  the  foci  lie 
midway  between  the  centre  and  the  vertices  ;  if  the  vertices  lie  midway  between 
the  centre  and  the  foci. 

14.  Show  that  the  tangents  at  the  ends  of  either  axis  of  a  central  conic  are 
parallel  to  the  other  axis ;  and  also  that  tangents  at  the  ends  of  any  chord 
through  the  centre  are  parallel. 

16.  Find  the  equations  of  the  tangents  and  normals  at  the  ends  of  the  latera 
recta.     Where  do  they  meet  the  a;-axis  ?  One  Ans.  y  +  ex  =  a. 

/    16.   Show  that  the  line  ?/  =  2  a;  —  y'|  touches  the  conic 

3  x2  -  6  2/2  =  1. 

17.  Find  the  equations  of  the  tangents  to  the  ellipse  x"^  +  4ty'^  =  \Q  which 
make  angles  of  45°  and  60°  with  the  x-axis. 

18.  Show  that  the  directrix  is  the  polar  of  the  focus. 

19.  If  the  slope  of  a  moving  line  remains  constant,  the  locus  of  its  pole  with 
respect  to  a  central  conic  is  a  straight  line  through  the  centre  of  the  conic. 


110] 


THE   ELLIPSE  AND   HYPERBOLA 


155 


110.    Conjugate  Hyperbolas. 

The   two   hyperbolas   whose 
equations  are 


^_r_i 

a'     W~    ' 


and 


or 


x" 


-1, 


62 


-".=1, 


(1) 


(2) 


are  so  related  that  the  trans- 
verse axis  of  the  one  is  the 
conjugate  axis  of  the  other. 

The  two  hyperbolas  are  then 
said  to  be  conjugate  to  one 
another. 


The  eccentricity  of  the   Conjugate  Hyperbola*  is  ei  = 


the  coordinates  of  its  foci  are  (0,  ±  be^) ;  the  equations  of  its  direc- 
trices are  2/  =  ±  — ;  and  its  latus  rectum  is  — ^• 

When  a  =  b,  equations  (1)  and  (2)  become,  respectively, 


V6'  +  a' 


/2 


=  a\\ 


and  y^  —  Qi?  =  c?.) 

Hence  if  a  hyperbola  is  equilateral  or  rectangular  [§  90,  (15)], 
its  conjugate  is  also  rectangular. 

Two  conjugate  hyperbolas  are  not,  in  general,  similar  (§  116),  i.e. 
of  the  same  shape,  but  two  conjugate  rectangular  hyperbolas  are 
equal." 

*  The  hyperbola  (2)  is  usually  called  the  Conjugate  Hyperbola,  while  (1)  is  called 
the  Original,  or  Primary  Hyperbola.  It  is  to  be  noticed  that  the  equation  of  the 
conjugate  hyperbola  is  found  by  changing  the  sign  of  one  member  ot  the  equation  ol 
the  primary  hyperbola.  Likewise  the  equation  of  the  conjugate  ellipse  is  found  to  be 

Henoe  the  conjugate  of  an  ellipse  is  imaginary. 


156  TPIE   ELLIPSE  AND   HYPERBOLA  [111 

111.    To  find  the  locus  of  the  point  of  intersection  of  two  perpendicu- 
lar tangents  to  the  conic 

or     ¥ 
The  equation  of  any  tangent  to  (1)  may  be  written  (§  108) 


y  =  mx  +  ^ahn^  ±  h^.  (2) 

If  this  line  (2)  passes  through  (ccj,  2/i)>  we  shall  have 


which  when  rationalized  becomes 

(x,'-a')m'-2x,y,7n+(y,'Tb')=0.  (3) 

This  equation  is  a  quadratic  in  m  whose  two  roots  are  the  slopes 
of  the  two  tangents  which  pass  through  the  point  (x^,  y^),  whose 
locus  is  required. 

Let  mi  and  mg  be  the  two  roots  of  (3)  ;  then  (§  68) 

mimg  —  — • 

Xi  —  a^ 

The  two  tangents  will  be  at  right  angles  if  mimg  =  —  1  (§  45) ; 
i.e.  if 

or  ,  aj/  +  2/i'  =  a'±&2.  (4) 

The  required  locus  is,  therefore,  the  circle 

x^  +  y^  =  a^±b^,  (5) 

which  is  called  the  Director  Circle  of  the  conic. 

Cor.  I.     Ifa<b,  the  director  circle  of  a  hyperbola  is  imaginary. 
Hence  one  of  the  director  circles  of  two  conjugate  hyperbolas  is  always 
imaginary. 

Cor.  II.     The  director  circle  of  the  ellipse  ^  +  ^  =  1  passes  through 
x^     7/2  a^     b^ 

the  foci  of  the  hyperbolas  ——^  =  ±1  and  vice  versa, 
a^     b^ 

What  does  this  mean  when  a  =  b? 


112] 


THE   ELLIPSE   AND   HYPERBOLA 


157 


112.   Auxiliary  Circle,  and  Eccentric  Angle. 

I.     The  circle  described  on  the  major  axis  of  an  ellipse  as  diame- 
ter is  called  the  Auxiliary  Circle. 

Y 


If  the  equation  of  the  ellipse  is 


(1) 


the  equation  of  the  auxiliary  circle  will  be 

x^  +  f  =  a\  (2) 

If  the  ordinate  NP  of  any  point  P  on  the  ellipse  is  produced  to 
meet  the  auxiliary  circle  in  Q,  then  P  and  Q  are  called  Corresponding 
Points. 

Let  P(.Ti,  .?/i)  and  Q{x^,  y^  be  any  two  corresponding  points ;  then, 
since  these  points  are  on  (1)  and  (2),  respectively, 

2/1  =  5  Va^  -  xi'^  (3) 

a 

and  ?/2  =  Va^  —  x^^-  (4) 

(5) 


2/2  =  Va^  —  x^. 

...  yi=K 

That  is,  the  ordinates  of  corresponding  points  are  in  a  constant  ratio. 
Ex.     Show  that  the  area  of  the  ellipse  is  irab. 


158 


THE  ELLIPSE  AND  HYPERBOLA 


[112 


The  angle  XOQ  is  called  the  Eccentric  Angle  of  the  point  P.     It 
will  be  denoted  by  <^. 
Then  the  coordinates  of  the  point  Q  are 

a^  =  a  cos  (f>j    y2  =  ci  sin  <^. 

Since  2/1  =  -2/2  =  &  sin  <f>,  the  coordinates  of  P  are 
a 

Xi  =  a  cos  «t),    Vi-h  sin  <|>.  (6) 

II.   The  circle  described  on  the  transverse  axis  of  a  hyperbola  as 
diameter  may  be  called  the  Auxiliary  Circle  of  the  hyperbola. 


Let  P(x,  y)  be  any  point  on  the  hyperbola  and  NP  its  ordinate. 
Draw  NQ  tangent  to  the  auxiliary  circle  at  Q,  so  that  P  and  Q  are 
on  the  same  side  of  the  transverse  axis  when  P  is  on  the  right 
branch,  and  on  opposite  sides  when  P  is  on  the  left  branch  of  the 
curve.  Then,  as  P  describes  the  complete  hyperbola  in  the  direc- 
tion indicated  by  the  arrows,  Q  will  move  consecutively  around  the 
circle  in  the  direction  indicated.  Thus,  for  every  position  of  P  on 
the  hyperbola,  there  is  one  and  only  one  corresponding  position  of 
Q  on  the  circle. 

Hence  P  and  Q  may  be  called  Corresponding  Points,  and  the 
angle  XOQ  =  <fi  may  be  called  the  Eccentric  Angle  of  the  point  P. 


11^  THE  ELLIPSE  AND  HYPERBOLA  159 

Let  the  equation  of  the  hyperbola  be 

$-$='■  ,  (^ 

Then  ON=  a;  =  a  sec  +,  _  (^) 

which  substituted  in  (7)  gives 

y  =  b  tan  ^.  (9) 

That  is,  P  is  the  point  (a  sec  «|>,  b  tan  <|>). 

Similarly,  se^  -\-  y^  —  b^  is  the  auxiliary  circle  of  the  conjugate 
hyperbola,  and  (a  tan  <f),  b  sec  <f>)  is  any  point  on  the  curve  if  <f>  is 
measured  clockwise  from  the  positive  end  of  the  2/-axis ;  if  <^  is  meas- 
ured from  the  x-axis  the  point  is  (a  cot  <^,  b  esc  <^). 

113.  To  find  the  equatioii  of  the  straight  line  joining  two  points  on  a 
conic  whose  eccentric  angles  are  <^  and  <^'. 

If  the  conic  is  an  ellipse,  the  points  are  (§  112) 

(a  cos  <^,  b  sin  <^)     and     (a  cos  <^',  b  sin  <^'). 

The  equation  of  the  line  through  these  points  is  [(3),  §  44] 

a;  —  a  cos  </>       _       y  —  6  sin  <^  ^^x 

a  cos  <fi  —  a  cos  <f>'      b  sin  <f>  —  b  sin  <f>' 

Since     cos  </>  —  cos  <^'  =  —  2  sin  |(</>  +  <^')  sin  ^{<f>  —  <t>') 

and  sin  <^  —  sin  <^'  =  2  cos  J(<^  +  <^')  sin  ^(<^  —  <^'), 

equation  (1)  reduces  to 

^-"^^^         ^       |-^^^^  (2) 

-2sinK</»  +  <^')      2  cos  K<^ +  </>')* 

.-.   ^  cos  i(<|>  + 4.0  + 1  sin  !(<!>+ <!»')  =  cos  |(<|»-<|»0,  (3) 

which  is  the  required  equation. 

In  like  manner  the  equation  of  the  line  joining  the  points  (a  sec  <f>y 
b  tan  <^)  and  (a  sec  <^',  b  tan  tf>')  on  the  hyperbola  can  be  shown  to  be 

^  cos  ^(4.  -  4>')  - 1  sin  |(<|>  +  4>')  =  cos  1(4.  +  4»').  W 


160  THE   ELLIPSE   AND   HYPERBOLA  [114 

To  find  the  equation  of  the  tangent  at  the  point  (}>,  we  put  <^'  =  <^ 
in  equations  (3)  and  (4),  and  we  obtain  for  the  ellipse 

^cosc|»  +  |sm<|>  =  l,  (5) 

and  for  the  hyperbola 

-sec<|>-gtan<|»  =  l.  (6) 

From  equation  (3)  we  see  that  if  the  sum  of  the  eccentric  angles 
of  two  points  on  an  ellipse  is  constant  and  equal  to  2  a,  the  equation 
of  the  line  joining  them  is 

-  cos  «  +  T  sin  a  =  cos  J(</>  —  <^').  (7) 

Hence  the  chord  is  always  parallel  to  the  tangent 


i 


cos  a  + 1  sin  «  =  1.  (8) 

Conversely,  in  a  system  of  parallel  chords  of  an  ellipse,  the  sum 
of  the  eccentric  angles  of  the  extremities  of  any  chord  is  constant. 

Similarly  from  equation  (4)  we  see  that  if  the  sum  of  the  eccen- 
tric angles  of  two  points  on  a  hyperbola  is  constant  and  equal  to  2  a, 
the  equation  of  the  chord  through  these  points  is 

-  cos  i(<f>  —  <^')—  I  sin  a  =  cos  a,  (9) 

and  therefore  the  chord,  and  the  tangent  at  the  point  a,  viz., 

*t  sin  a  —  cos  a,  (10) 

ah  '  ^ 

always  meet  the  2/-axis  in  the  same  fixed  point. 

114.  To  find  the  equation  of  the  normal  at  any  point  in  terms  of  the 
eccentHc  omgle  of  the  point. 

Let  (a  cos  <f),  b  sin  <^)  (§  112)  be  any  point  on  the  ellipse;  then  the 

slope  of  the  tangent  at  the  point  <^  is  -  ^£2^.     [§  113  (5).-j 

a  sin  </) 


114]  THE   ELLIPSE  AND  HYPERBOLA  161 

Hence  the  equation  of  the  normal  at  <^  is  [(2),  §  62] 

,    .     ,      a  sin  <^  ,  , .  .^  ^ 

y-b8m<l>  =  :^-^^(x-acos4>),  (1) 

Similarly  we  find  the  equation  of  the  normal  to  the  hyperbola  at 
the  point  (a  sec  <f>,  b  tan  </»)  to  be 


EXAMPLES 


1.  The  point  P(—  3,  —  1)  is  on  the  ellipse  x^  +  S  y"^  =  \2  ;  find  the  correspond- 
ing point  on  the  auxiliary  circle,  and  the  eccentric  angle  of  P. 

2.  An  ellipse  slides  between  two  perpendicular  lines  ;  show  that  the  locus  of 
the  centre  is  a  circle.     (§  111.) 

3.  Show  that,  for  all  values  of  6,  tangents  to  the  ellipse  -^  +  p  =  -^  *^  points 
having  the  same  abscissa  meet  the  oj-axis  in  the  same  point.  Hence  show  how  a 
tangent  can  be  drawn  to  an  ellipse  from  any  point  on  the  x-axis. 

4.  Two  tangents  are  drawn  to  a  conic  from  any  point  on  the  director  circle  ; 
prove  that  the  sum  of  the  squares  of  the  chords  which  the  auxiliary  circle  inter- 
cepts on  them  is  equal  to  the  square  of  the  line  joining  the  foci.     (See  (9),  §  109.) 

5.  If  the  points  Q  and  Q'  are  taken  on  the  minor  axis  of  a  conic  such  that 
Q0=  OQ'  =  OF,  where  0  is  the  centre  and  F  a  focus,  show  that  the  sum  of  the 
squares  of  the  perpendiculars  from  Q  and  Q'  on  any  tangent  to  the  conic  is  con- 
stant. 

6.  A  line  is  drawn  through  the  centre  of  a  conic  parallel  to  the  focal  radius/ 
of  a  point  P  and  meeting  the  tangent  at  P  in  Q.     Find  the  locus  of  Q. 

7.  From  one  focus  of  an  ellipse  a  perpendicular  is  drawn  to  any  tangent  and 
produced  to  an  equal  distance  on  the  other  side.  Show  that  its  terminus  Q  is  in 
the  straight  line  through  the  other  focus  and  the  point  of  tangency.  Also  find 
the  locus  of  Q. 

8.  Show  that  the  locus  of  the  point  of  intersection  of  tangents  to  an  ellipse  at 
two  points  whose  eccentric  angles  differ  by  a  constant  is  an  ellipse. 

x' 

[If  the  tangents  at  0  +  a  and  0  —  a  meet  at  (x',  y')>  then  —  =  cos  0  sec  a, 

^  =  sin  0  sec  a.    Eliminate  0  for  the  locus.] 

What  is  the  corresponding  theorem  for  the  hyperbola  ? 


A 


(1) 


162  THE   ELLIPSE   AND  HYPERBOLA  [115 

115.  Def.  An  Asymptote*  to  a  curve  is  the  limiting  position 
of  the  tangent  line  as  the  point  of  contact  moves  off  to  an  infinite 
distance,  while  the  line  itself  remains  at  a  finite  distance  from  the 
origin. 

Tojind  the  asymptotes  of  the  hyperbola. 

a^     b'" 

As  in  §  108,  the  abscissas  of  the  points  where  the  line 

y  =  mx  4-  c  (2) 

meets  the  hyperbola  are  given  by  the  equation 

x"  (aV  -  5')  +  2  a'cmx  +  a'  (c"  +  b^)  =  0.  (3) 

If  the  line  (2)  becomes  an  asymptote,  both  roots  of  equation  (3) 
must  become  infinite.  Hence  the  coefficients  of  a^  and  x  must  both 
approach  zero  (§  77).     That  is, 

a^cm  =  0,     and     a^m^  —  b^  =  0. 

.-.    lim  c  =  0,   and   lim  m  =  ±  —  (4) 

a  ^  ^ 

Substituting  these  limiting  values  in  (2),  we  have  for  the  required 
equations  of  the  asymptotes 

or  expressed  in  one  equation  \)  C^ 

Therefore  the  hyperbola  has  two  asymptotes,  both  passing  through 
the  centre  and  equally  inclined  to  the  transverse  axis. 

The  equations  of  the  asymptotes  to  a  hyperbola  can  also  be  found 
by  considering  the  limiting  form  of  the  equation  of  the  tangent  as  the 
point  of  contact  moves  off  to  an  infinite  distance. 

The  equation  of  the  tangent  to  (1)  at  (x'j  y')  is 

xx'     vv'      .  ,_ 

^-f  =  l-  ■  (7) 

*  Greek,  dtriJ/xn-Twros,  not  falling  together. 


115]  THE  ELLIPSE   AND   HYPERBOLA  163 

Since  the  point  (x\  y')  is  on  the  conic  (1),  we  have 


0/ 

Hence  quotation  (7)  may  be  written  0  H  _ 

If  now  the  point  of  contact  («',  y')  moves  off  to  an  infinite  distance 

so  that  x'  becomes  infinite,  the  limiting  position  of  the  line  (8)  is 

given  by  the  equation  x     v 

5±f  =  0,  (9) 

which  is  the  same  as  equation  (5)  above. 

CoR.  I.  Two  conjugate  hyperbolas  have  the  same  asymptotes,  which 
are  the  diagonals  of  the  rectangle  formed  by  the  tangents  at  their  vertices. 

CoR.  II.  A  straight  line  parallel  to  an  asymptote  will  meet  the  conic 
in  one  point  at  infinity. 

For,  if  c  is  not  zero,  only  one  root  of  (3)  is  infinite. 
Cor.  III.     The  line  y  =  mx  will  cut  the  hyperbola  in  real  or  imagi- 
nary points  according  as  m<,or^--     It  will  meet  either  the  hyperbola 

a 

or  its  conjugate  in  real  points  for  all  values  of  m. 

Cor.  IV.     The  asymptotes  of  an  ellipse  are  imaginary. 

For,  if  we  change  the  sign  of  6^,  the  values  of  m  for  infinite  roots 
in  (3)  become  imaginary. 

It  is  to  be  noticed  that  the  equations  of  two  conjugate  hyperbolas 
and  the  equation  of  their  common  asymptotes,  viz., 

-2-^=±l  and  ^-^  =  0, 
a^     Ir  a^     Ir 

differ  only  in  their  constant  terms.  Moreover,  this  must  always  be 
true ;  for  any  transformation  of  coordinates  will  affect  the  first  mem- 
bers of  these  equations  in  precisely  the  same  way.  Hence  the  new 
equations  will  differ  only  in  their  constant  terms  (not  usually  by 
unity) ;  and  the  value  of  the  constant  in  the  equation  of  the  asymp- 
totes will  be  equal  to  half  the  sum  of  the  constants  in  the  equations 
of  the  two  hyperbolas. 


164  THE  ELLIPSE  AND  HYPERBOLA  [116 

116.   Similar  and  Coaxial  Conies. 

Since  a^Ksbnd  6^^  are  the  semi-axes  of  the  ellipse 

S+P=^'  (1) 

its  eccentricity  is  given  by  the  equation 


e  = 


(§  107.) 


a^K  a 

That  is,  the  eccentricity  of  (1)  is  the  same  as  the  eccentricity  of 
the  ellipse  represented  by  the  standard  equation 

5  +  S  =  l-  (2) 

Two  conies  having  the  same  eccentricity  are  said  to  be  similar ;  for 
one  is  then  merely  a  magnification  of  the  other. 

Conies  having  their  axes  on  the  same  lines  are  said  to  be  Coaxial. 

Hence  if  K  is  an  arbitrary  parameter,  (1)  will  represent  a  system 
of  similar  and  coaxial  ellipses. 

For  any  particular  value  of  K  the  equations 

i-t=±K  (3) 

represent  a  pair  of  conjugate  hyperbolas  (§  110). 

If,  however,  jfiTis  arbitrary,  equations  (3)  will  give  (as  in  the  case 
of  the  ellipse)  a  system  of  similar  and  coaxial  hyperbolas,  together 
with  their  corresponding  conjugate  hyperbolas,  which  are  also  similar. 
It  follows  from  §  115  that  these  two  infinite  systems  of  hyperbolas 
all  have  the  same  asymptotes.  Moreover,  the  asymptotes  are  the 
limit  which  both  systems  approach  as  K  becomes  zero.  Thus  two 
intersecting  lines  are  not  only  one  of  a  system  of  similar  and  coaxial 
hyperbolas,  but  may  also  be  regarded  as  a  pair  of  self-conjugate  hyper- 
bolas. 

It  is  also  to  be  noticed  that  although  both  axes  of  two  intersecting 
lines  are  zero,  the  limit  of  their  ratio  as  they  approach  zero  is  the 
tangent  of  half  the  angle  between  the  lines. 


Cor.     The  axes  of  similar  conies  are  proportional. 


117] 


THE  ELLIPSE  AND  HYPERBOLA 


165 


117.    To  find  the  locus  of  the  middle  points  of  a  system  of  parallel 
chords  of  a  central  conic. 


I.  Let  AB  be  any  one  of  a  system  of  parallel  chords  of  the 
ellipse 

^  +  t  =  K. 


(1) 


Let  P(x',  y')  be  the  middle  point  of  ABj  and  y  its   inclination 
to  the  aj-axis. 

Then  the  equation  of  AB  may  be  written  [§  43,  (4)] 

x—x'     y  —  y' 

cos  y        Sin  y 

ot  x  =  x'  -\-r  cos  y,    y  =  y'  -;  r  sin  y,  (2) 

where  r  is  the  distance  from  (a;',  y')  to  any  point  (a;,  y)  on  the  line. 

If  the  point  (x,  y)  is  on  the  ellipse,  these  values  (2)  may  be  sub- 
stituted in  equation  (1) ;  this  gives 

(x'  +  r  cos  y^*^  ,  (7'  -f  r  sin  y)'^      j^  ^^ 
-p  ~ 7^ "-^  =  -fl.,  or 


/cos 


^2       ^        ^2     y       T-     ^        ^2  -^  52        y     ^^2-^52         ^         "•      W 

The  values  of  r  lound  by  solving  this  quadratic  equation  are 
the  lengths  of  tka  lines  PA  and  PB,  which  can  be  drawn  from  P 


166  THE  ELLIPSE  AND  HYPERBOLA  [117 

along  AB  to  the  ellipse.  Since  P  is  the  middle  point  of  the  chord, 
these  two  values  of  r  must  be  equal  in  magnitude  and  opposite  m 
sign  ;  i.e.  the  sum  of  the  roots  of  (3)  must  be  zero.     Hence  (§  68) 

a;'cos  Y     y'  siny^r.  .^. 

a'      ^      b'  '  ^  ^ 

The  required  locus  is,  therefore,  the  straight  line 

y= cot  7  •  a?.  (5) 

Henee  every  diameter  (§  99)  of  an  ellipse  passes  through  the 
centre. 

CoR.  I.  All  chords  intercepted  on  the  same  line,  or  on  a  series  of 
parallel  lines,  by  a  system  of  similar  and  coaxial  ellipses  are  bisected  by 
the  same  diameter. 

Since  equation  (5)  is  independent  of  K,  the  locus  of  P  is  the 
same  whatever  value  may  be  given  to  K  in  (1).     (§  116.) 

CoR.  II.  If  a  straight  line  meets  each  of  two  similar  and  coaxial 
ellipses  in  two  real  points,  the  tic o  portions  of  the  line  intercepted  between 
them  are  equal ;  i.e.  AA  =  BB'. 

CoR.  III.  Chords  of  an  ellipse  which  are  tangent  to  a  similar  and 
coaxial  ellipse  are  bisected  at  the  point  of  contact. 

CoR.  IV.  The  tangent  at  either  extremity  of  any  diameter  is  parallel 
to  the  chords  bisected  by  that  diameter. 

II.  In  like  manner,  if  "y  is  the  inclination  to  the  a>-axis  of  a 
system  of  parallel  chords  of  the  hyperbolas 

a^     b^  '  ^  ^ 

we  find  the  locus  of  the  middle  points  of  the  chords  to  be  the 
straight  line 

y=-^cot7-a^,  (7) 

for  all  values  of  K,  including  the  case  K—  0. 

Hence  all  diameters  of  a  hyperbola  pass  through  the  centre. 

The  preceding  corollaries  apply  also  to  similar  and  coaxial  hyper- 
bolas. 


117] 


THE  ELLIPSE   AND  HYPERBOLA 


167 


Cor.  V.  Cliords  intercepted  on  the  same  line,  or  on  a  system  ofpar- 
ellel  lines,  by  two  conjugate  hyperbolas,  and  their  asymptotes,  are  bisected 
by  the  same  diameter. 

Cor.  YI.  If  a  straight  line  meets  each  of  two  conjugate  hyperbolas 
in  real  points,  the  two  poHions  of  the  line  intercepted  between  the  curves 
are  equal.  TJie  portions  intercepted  between  either  hyj)erbola  and  the 
asymptotes  are  also  equal;  i.e.  A" A  =  BB"  and  A' A  =  BB'.  Hence 
the  part  of  a  tangent  to  a  hyperbola  included  between  the  two  branches 
of  its  conjugate,  and  also  the  part  included  between  its  asymptotes,  are 
bisected  at  the  point  of  contact. 

Ex.  1.  Find  the  locus  of  the  middle  points  of  chords  of  the  ellipse 
4  x2  +  9  y2  =  36  parallel  to  3  a;  -  2  ?/  =  1. 

Ex.  2.  Find  the  equation  of  the  chord  of  the  hjrperbola  26  x^  -  16  j/2  =  400 
which  is  bisected  at  the  point  (2,  -  6). 

Ex.  3.  Find  the  equation  of  the  chord  of  the  ellipse  4  a;^  +  8  t/^  =  32  which 
is  bisected  at  the  point  (  —  2,  1). 


\ 


A.l^L/ 


168  THE  ELLIPSE  AND   HYPERBOLA  [118 


Conjugate  Diameters 

118.  We  have  seen  in  §  117  that  all  diameters  of  a  central  conic 
pass  through  the  centre.  Conversely,  every  chord  which  passes 
through  the  centre  is  a  diameter,  i.e.  bisects  some  system  of  parallel 
chords.  For,  by  giving  y  a  suitable  value,  equations  (5)  and  (7)  of 
§  117  may  be  made  to  represent  any  chord  through  the  centre. 

If  y'  is  the  inclination  to  the  aj-axis  of  the  diameter  which  bisects 
all  chords  whose  inclination  is  y,  we  have,  from  (5)  and  (7)  of  §  117, 

h- 
tan  y'—^—  cot  y, 
a^ 

or  tan  y  tan  y'  =  q=  (1) 

a^ 

Let  y  =  mx  and  y  =  m'xhe  any  two  diameters. 
Then,  if  the  first  bisects  all  chords  parallel  to  the  second,  we  have 
from  (1)  ,  o 

mm'=T^'  '    (2) 

Since  this  is  the  only  condition  that  must  hold  in  order  that  the 
second  may  bisect  all  chords  parallel  to  the  first,  it  follows  that, 
if  one  diameter  of  a  conic  bisects  all  chords  parallel  to  a  second,  the 
second  diameter  will  also  bisect  all  chords  parallel  to  the  first. 

Def.  Two  diameters,  so  related  that  each  bisects  every  chord 
parallel  to  the  other,  are  called  Conjugate  Diameters.* 

For  example,  the  axes  are  a  pair  of  conjugate  diameters. 

From  equation  (2)  we  see  that  the  slopes  of  two  conjugate 
diameters  of  an  ellipse  have  opposite  signs,  whereas  in  the  hyper- 
bola the  signs  are  the  same.     (See  figures  under  §  117.) 

If  m  <  -,  then  m'  >  -,  numerically, 
a  a 

Hence  conjugate  diameters  of  an  ellipse  are  separated  by  the 
axes,  and  also  by  the  lines  ay=  ±bx;  while  conjugate  diameters  of 
a  hyperbola  are  separated  by  the  asymptotes,  but  not  by  the  axes. 

*  It  is  evident  that  none  but  central  conies  can  have  conjugate  diameters,  since  in 
the  parabola  all  diameters  have  the  same  direction  (§  99) . 


118]  THE  ELLIPSE  AND   HYPERBOLA  169 

If  m  =  - ,  then  m'  = in  the  ellipse. 

The  two  diameters  are  then  equally  inclined  to  the  major  axis,  and, 

from  the  symmetry  of  the  curve,  the  two  diameters  are  equal  in  length. 

The  equations  of  the  equal  conjugate  diameters  of  an   ellipse  are, 

therefore,  i, 

v=±l^.  (3) 

If  m  =  ±  -,  then  in  the  hyperbola m'  =  ±  -,  respectively. 
a  a 

Therefore  equi-con jugate  diameters  of  a  hyperbola  coincide  with 
an  asymptote,  so  that  an  asymptote  may  be  regarded  as  a  self-conju- 
gate diameter. 

The  equi-con  jugate  diameters  of  a  conic,  therefore,  in  all  cases 
-V coincide  in  direction  with  the  diagonals  of  the  rectangle  formed  by 
>a:  tangents  at  the  ends  of  its  axes. 

^      CoR.  I.   If  tivo  diameters  are  conjugate  with  respect  to  one  of  two  con- 

^"  jugate  hyperbolas,  they  will  be  conjugate  with  respect  to  the  other  also. 

^%m.  and  (7),  §  117.] 

v^  >^5^     CoR.  II.    One  of  two  conjugate  diameters  of  a  hyperbola  meets  the 

y     *  jf  curve  in  real  points,  and  the  other  meets  the  conjugate  hyperbola  in  real 

JTT^  points.     (Cor.  Ill,  §  115.) 

r  ^^^     For  this  reason  we  will  call  the  extremities  of  any  diameter  of  a 
^  *^  ^  hyperbola  the  points  in  which  it  cuts  either  the  primary  or  the  con- 
K^J  jugate  hyperbola,  as  the  case  may  be ;  and  the  length  of  the  diameter 

^  O  will  be  the  distance  between  these  points. 
^^Jv       Cor.  III.    Tangents  at  the  ends  of  any  diameter  are  paraMel  to  the 
^  n   conjugate  diameter.  /N^ 

^%         _ ^'^^^ 


•1 


^        Ex.  1.   Write  down  the  equations  of  the  diameters  conjugate  to    f^      \iJi*^ 
^  x-y  =  0,x-\-y  =  0,by  =  ax,ay  =  bx.  ^^  ^ 


Ex.  2.   In  the  ellipse  2x^  +  ^y^  =  S,  find  two  conjugate  diameters,  one  of 
which  bisects  the  chord  as  +  2  y  =  2. 
g^^^^  Ex.  3.   Find  the  equation  of  the  diameter  of  the  hyperbola  16x^  —  9y^  =  144 
TC      conjugate  to  x  -f-  2  y  =  0. 
•^>i^    y^  ^^*  "*•   ^^^^  ^^^  conjugate  diameters  of  the  ellipse  4  a?  -f-  25  y*  =  100,  one  of 
/which  passes  through  the  point  (3,-1). 
'^^.A        Ex.  6.   Find  the  equation  of  the  chord  of  the  hyperbola  x^  —  y^  =  l6,  whose 
^«^ddle  point  is  (- 2, 


'N^ftW-'"^ 


170 


THE  ELLIPSE  AND  HYPERBOLA 


[119 


119.   Oiven  the  extremity  of  any  diameter ,  to  find  the  extremities  of 
the  conjugate  diameter. 


I.   Let  Pi(fl7i,  2/i)j  -Pa  (''^2)  2/2)  be  the  extremities  of  two  conjugate 
diameters  of  an  ellipse. 

Then  the  equations  of  OP^  and  OP2  are 


or 


=  —x  and  y  =  ^x. 

Xi  X.2 


XiX.2 

.2  ~^ 


a^ 
=  0. 


[(1),§118.] 


(1) 


a^    ■    h^ 

Let  <^i,  <^2  be  the  eccentric  angles  of  P^  Pg,  respectively. 
Then  ajj  =  a  cos  <^i,        2/i  =  ^  sin  <^i, 

0^2  =  a  cos  <^2>        2/2  =  ^  sin  <^2-  (§  112, 1.) 

Substituting  these  values  in  (1),  we  have 

cos  <^i  cos  </)2  +  sin  <^i  sin  <^2  =  cos  (<^i  ~  <^2)  =  0.  (2) 

.*.     ^1  -"^  <j^2  ^^^  90  ./ 

That  is,  the  eccentric  angles  of  the  extremities  of  two  conjugate 
diameters  of  an  ellipse  differ  by  a  right  angle.  >Hence  the  corre- 
sponding diameters  OQi,  OQ2  of  the  auxiliary  circle  are  perpendicu- 
lar to  one  another. 


119] 
Since 


THE  ELLIPSE  AND  HYPERBOLA 


171 


<^2  =  <^i±90°, 
sin  <;^2  =  ±  cos  <^i,         cos  <f>2  —  T  sin  <^i. 
Therefore  the  extremities  of  two  conjugate  diameters  of  an  ellipse 
may  be  written 

JPi (a cos <|)i,  6 sin <|»i)  and  1*2 ( ^  « siii<|>i,  ±  6 cos <|>i) , '] 

(3) 


or 


^i(a?i,  Vi)  and  Pgf  T  ^2/1?  ±  -^ij- 


(4) 


II.   If  Pi,  F2  are  the  extremities  of  two  conjugate  diameters  of  a 
hyperbola,  equation  (1)  becomes 

Then  from  §  112,  II,  and  §  118,  Cor.  II,  we  also  have 
Xi  =  a  sec  ^1,    2/1  =  6  tan  <^i, 
X2  =  a  tan  <}>2,    2/2  =  6  sec  ^2- 
Substituting  these  values  in  (4)  gives 

sec  <^i  tan  <f>2  —  tan  <^i  sec  <^2  =  0, 
sin  <f>2  sin  <^i 


or 


cos  ^1  cos  <^2     cos  <^i  cos  <f>2 
,;  ^2  =  <^  or  <^2  =  TT  —  <j!>i 


=  0. 


(6) 
(7) 


172  THE  ELLIPSE  AND  HYPERBOLA  [120 

That  is,  the  eccentric  angles  of  the  ends  of  two  conjugate  diame- 
ters of  a  hyperbola  are  either  equal  or  supplementary.  Therefore 
the  corresponding  diameters  OQi,  OQ2  of  the  auxiliary  circles  are 
equally  inclined  to  the  transverse  axes  of  the  two  conjugate  hyper- 
bolas. 

Since  tan  <^2  =  ±  tan  <^i  and  sec  <^2  =  ±  sec  «^i, 

the  extremities  of  any  two  conjugate  diameters  of  a  hyperbola  may 
be  expressed  in  the  form 

Pi  (a  sec  <|>i,  b  tan  4>i)  and  J*2  ( ±  « tan  c|>i,  ±  b  sec  «|>i), 
Pi(^  2/1)  and  J^2(±^yi,  ±-^1)' 


or 


(8) 


120.  The  sum  of  the  squares  of  two  conjugate  semi-diameters  of  an 
ellipse  is  constant. 

Let  the  extremities  of  any  two  conjugate  diameters  be  [§  119,  (3)] 

Pi  (a  cos  </),  h  sin  <^)  and  P2(T  a  sin  <^,  ±h  cos  <^). 

Let  OPi  =  a',  OP2  =  b'j  0  being  the  centre. 

Then  a'^  =  a'  cos^  <f>-}-b^  sin^  <^,  [(4),  §  7] 

h"'  =  a^sm^<f>-\-b^cos^<f>. 

...  a'^  +  6^2  =  a2  +  62. 

121.  The  area  of  the  parallelogram  formed  by  tangents  at  the  ends 
of  conjugate  diameters  of  an  ellipse  is  constant. 

Let        Pj  (a  cos  <^,  b  sin  <^)  and  P2  ( =F  «  sin  <l>,  ±b  cos  <^) 

be  the  extremities  of  any  two  conjugate  diameters,  and  let  ABCD  be 
the  parallelogram  formed  by  tangents  at  the  ends  of  these  diameters. 
Draw  P^N  perpendicular  to  OP2 ;  then 

Area  ABCB  =  4.0P2'  P^N=  4  6' .  P,N. 

Since  OP2  is  parallel  to  the  tangent  at  Pj  [§  118,  Cor.  Ill],  the 
equation  of  OP2  may  be  written  [(5),  §  113] 

-  cos  A  4- 1  sin  d>  =  0.  X 

a  0  mm      \ 


121] 


P,N= 


THE  ELLIPSE   AND  HYPERBOLA 

\ 

cos^  <^  +  sin^  <^  ah      * 


173 


ab 


Vcos^ 
-5 


52 
.-.  Area  ABCD  =  ^ab. 
Y 


Cor.    If  angle  P1OP2  =  w,  then 


-x;^ 


^ 


a'       a'V 


EXAMPLES 


1.  The  difference  of  the  squares  of  two  conjugate  semi-diameters  of  a  hyper- 
bola is  constant. 

2.  The  area  of  the  parallelogram  formed  by  tangents  to  two  conjugate  hyper- 
bolas at  the  ends  of  two  conjugate  diameters  is  equal  to  4  ab. 

3.  If  w  denotes  the  angle  between  two  conjugate  diameters  of  a  hyperbola, 


then  sin  w  = 


ab 
a'h' 


\f       4.   Show  that  the  acute  angle  between  two  conjugate  diameters  of  an  ellipse 
^is  least  when  the  diameters  are  equal. 

5.  Show  that  the  eccentric  angles  of  the  extremities  of  the  equi-conjugate 
diameters  of  an  ellipse  are  45°  and  136°. 

6.  Conjugate  diameters  of  a  rectangular  hyperbola  are  equal,  and  equally 
inclined  to  the  asymptotes. 

7.  Tangents  to  two  conjugate  hyperbolas  at  the  extremities  of  two  conjugate 
diameters  meet  on  the  asymptotes.     (See  Fig.  §  117,  H.) 


174  THE   ELLIPSE   AND   HYPERBOLA  [122 

8.  The  area  of  the  triangle  formed  by  two  conjugate  semi-diameters  and  the 
chord  joining  their  ends  is  constant. 

9.  Prove  that  for  all  values  of  m  the  line 


passes  through  the  extremities  of  two  conjugate  diameters  of  an  ellipse.     "What 
is  the  corresponding  equation  for  the  hyperbola  ? 

10.  The  product  of  the  focal  radii  of  a  point  P  is  equal  to  the  square  of  the 
semi-diameter  parallel  to  the  tangent  at  P. 

122.  To  find  the  eqvxition  of  a  hyperbola  when  referred  to  its  asymp- 
totes as  axes  of  coordinates. 

The  equation  of  the  asymptotes,  referred  to  themselves  as  axes  of 
{coordinates,  is  xy  =  0. 

Therefore  the  equations  of  any  two  conjugate  hyperbolas  referred 
to  them  is  of  the  form  (§  115) 

xy  =  ±K.  (9) 

Hence  the  equation  xy  =  K,  where  K  is  any  constant,  always 
represents  a  hyperbola  referred  to  its  asymptotes  as  axes  of  coordi- 
nates; so  that,  if  the  axes  of  coordinates  are  at  right  angles,  the 
hyperbola  xy  =  K  is  rectangular. 

123.  To  find  the  polar  equation  of  a  central  conic,  the  pole  being  ai 
the  centre. 

The  formulae  for  changing  from  rectangular  to  polar  coordinates 

are  (§6)  /i  •    z, 

^      ^  a;  =  pcos^,  2/  =  psm^. 

These  values  substituted  in 

or  n2=  ±^'^'  -  ±^'^' 

r 


a?  sin2  e±W  cos''  B     a"-  (a?  T  b"")  cos^  6 
which  is  the  required  equation. 


^      l-e2cos2e' 


123]  THE  ELLIPSE   AND   HYPERBOLA  175 

EXAMPLES  ON   CHAPTER  X 

1.  Show  that  the  sum  of  the  squares  of  the  reciprocals  of  two  perpendicular 
diameters  of  an  ellipse  is  constant.     (See  §  123.) 

2.  If  an  equilateral  triangle  is  inscribed  in  an  ellipse,  the  sum  of  the  squares 
of  the  reciprocals  of  the  diameters  parallel  to  the  sides  is  constant. 

3.  Find  the  inclination  to  the  major  axis  of  the  diameter  of  an  ellipse  the 
.sqii  ire  of  whose  length  is  (1)  the  arithmetical  mean,  (2)  the  geometrical  mean, 
and  (3)  the  harmonical  mean  between  the  squares  on  the  major  and  minor  axes. 
(§123.)  ^ws.  «o  (3),  45°. 

4.  The  locus  of  the  poles  of  a  series  of  parallel  chords  is  the  diameter  which 
bisects  the  chords.  Hence  the  line  joining  the  intersection  of  two  tangents  to 
the  centre  bisects  the  chord  of  contact. 

i<^     6.    Find    the    equations    of    two    conjugate   diameters  of    the   hyperbola 

b'^x^  —  a-y-  =  a-b'^,  one  of  which  bisects  the  chord  through  (0,  b)  and  (ae,  0). 

6.  In  the  hyperbola  4  aj2  _  5  ^-2  _  20  find  the  equations  of  two  conjugate 
diameters,  one  of  which  bisects  the  chord  2  a;  +  3  y  =  C. 

7.  If  straight  lines  drawn  through  any  point  of  an  ellipse  to  the  ends  of  any 
diameter  POP'  meet  the  conjugate  diameter  P\OP\'  in  Q  and  .B,  show  that 
Oq'OR  =  OPx^. 

8.  Show  that  the  locus  of  the  intersection  of  the  perpendiculars  from  the  foci 
upon  a  pair  of  conjugate  diameters  of  an  ellipse  is  a  similar  concentric  ellipse. 

9.  Two  conjugate  diameters  of  an  ellipse  are  drawn,  and  their  four  extremi- 
ties are  joined  to  any  point  on  a  given  circle  whose  centre  is  at  the  centre  of  the 
ellipse.     Show  that  the  sum  of  the  squares  of  these  four  lines  is  constant. 

10.  P,  is  a  point  on  a  branch  of  a  hyperbola,  P2  is  a  point  on  a  branch  of  its 
conjugate,  OPi  and  OP-i  being  conjugate  semi-diameters.  If  F\  and  F^  are  the 
interior  foci  of  these  two  branches^  respectively,  show  that 

F2P2  ~  FiPi  =  a'^b. 

11.  Find  the  equation  of  the  chord  passing  through  the  extremities  of  two 
conjugate  diameters. 

12.  The  lengths  of  the  chords  joining  the  extremities  of  two  conjugate 
diameters  of  an  ellipse  are 

Va2  +  6*  ±  a2g2  sin  2  <p. 

For  what  value  of  0  are  these  chords,  one  a  maximum  and  the  other  a  minimum  7 
Show  that  the  corresponding  result  for  the  hyperbola  is 

o«(8ec  0±tan0). 


176  THE  ELLIPSE   AND  HYPERBOLA  [123 

13.  Find  the  equations  and  the  coordinates  of  the  points  of  contact  of  tangents 
to  h^x^  ±  a^y^  =  a^^  which  make  equal  intercepts  on  the  axes. 

14.  If  the  normal  at  the  end  of  the  latus  rectum  of  an  ellipse  passes  through 
the  extremity  of  the  minor  axis,  show  that  the  eccentricity  is  given  by  the  equa- 
tion e*  -f  e2  =  1.  Find  the  corresponding  equation  for  the  hyperbola  and  inter- 
pret the  result. 

16.  If  any  ordinate  MP  oi  a  central  conic  is  produced  to  meet  the  tangent  at 
the  end  of  the  latus  rectum  through  the  focus  F  in  Q,  show  that  FP  =  MQ. 

16.  Find  the  product  of  the  segments  into  which  a  focal  chord  of  a  central 
conic  is  divided  by  the  focus. 

17.  Two  tangents  can  be  drawn  to  a  central  conic  from  any  point,  which  will 
be  real,  coincident,  or  imaginary  according  as  the  point  is  outside,  on,  or  inside 
the  conic.    Thus  determine  which  is  the  inside  of  a  hyperbola. 

18.  The  polar  of  a  point  P  with  respect  to  an  ellipse  cuts  the  minor  axis  in  A  ; 
and  the  perpendicular  from  P  to  its  polar  cuts  the  polar  in  B  and  the  minor  axis 
in  C.    Show  that  the  circle  through  A,  B,  and  C  will  pass  through  the  foci. 

[Prove  AO  '  OC=F'0'  OF,  where  O  is  the  centre.] 

19.  Prove  that  the  circle  on  any  focal  radius  as  diameter  touches  the  auxiliary 
circle. 

20.  Prove  that  the  line  lx  +  my  +  n  =  0  is  normal  to 

^2+52-1'    1^      Z2+„j2-  ^2  • 

[Compare  Zx  +  wiy  +  n  =  0  with  -^  -  -^  =  a'^-  b^.     (See  §  114.)] 

cos  0     sm  0  V         3         yj 

21.  Prove  that  a  circle  can  be  drawn  through  the  foci  of  a  hyperbola  and  the 
points  in  which  any  tangent  meets  the  tangents  at  the  vertices. 

22.  The  perpendicular  from  the  focus  of  an  ellipse  upon  any  tangent  and 
the  line  joining  the  centre  to  the  point  of  contact  meet  on  the  corresponding 
directrix. 

23.  If  Q  is  the  point  on  the  auxiliary  circle  corresponding  to  the  point  P  on 
the  ellipse,  the  normals  at  P  and  Q  will  meet  on  the  circle 

x2-\-y2  =  {a+by. 

24.  Prove  that  the  focal  radius  of  any  point  on  a  central  conic  and  the  per- 
pendicular from  the  centre  on  the  tangent  at  that  point  meet  on  a  circle  whose 
centre  is  the  focus  and  whose  radius  is  the  semi-major  axis. 


123]  THE  ELLIPSE  AND  HYPERBOLA  177 

25.  Show  that  the  minor  axis  is  a  mean  proportional  between  the  major  axis 
and  the  latus  rectum. 

26.  Any  tangent  to  an  ellipse  meets  the  director  circle  in  P  and  Q.  Prove 
that  OP  and  OQ  are  conjugate  diameters  of  the  ellipse. 

27.  Show  that  the  line  lx-{-my  =  n  will  touch 

^±|?  =  1  if  a2Z2±62m2  =  n2. 

The  line  a;  cos  a  +  y  sin  a  =  j)  will  touch  the  same  curves  if 
a2cos2a±52sin2a=p2. 

28.  Show  that  the  equation  of  the  locus  of  the  foot  of  the  perpendicular  from 
the  centre  of  a  conic  on  a  tangent  is  p2  =  q2  cos2  d±h^  sin2  d.    [Use  Ex.  27.] 

29.  If  a  polar  with  respect  to  a  central  conic  touches  the  circle  a;2  +  y2  _  52^ 
what  is  the  locus  of  the  pole  ? 

80.  Show  that  the  polar  of  any  point  on  either  of  the  curves 

a2^62 
with  respect  to  the  other  touches  the  first  curve. 

31.   The  polar  of  any  point  P  on  either  of  the  curves 

a;2     w2 

—  =  -1-1 

a2     62- ±  A 

with  respect  to  the  other  touches  the  first  curve  at  the  opposite  extremity  of  the 
diameter  through  P. 

82.  The  polars  of  any  point  with  respect  to  the  two  conies 

a2     62- ±^ 
are  parallel  and  equidistant  from  the  centre. 

33.  The  product  of  the  focal  radii  of  any  point  on  a  rectangular  hyperbola  is 
equal  to  the  square  of  the  distance  from  the  centre  to  that  point. 

34.  The  distance  of  any  point  Q  from  the  centre  of  a  rectangular  hyperbola 
varies  inversely  as  the  perpendicular  from  the  centre  upon  the  polar  of  Q. 

35.  If  the  normal  at  any  point  P  of  a  rectangular  hyperbola  meets  the  axes  in 
N  and  N*,  and  O  is  the  centre,  then  PN  =  PN'  =  OP. 

36.  A  line  parallel  to  the  y-axis  meets  two  conjugate  hyperbolas  and  one  of 
their  asymptotes  in  P,  Q^  P.  Show  that  the  normals  at  P,  Q,  B  meet  on  the 
OS-axis. 


178  THE  ELLIPSE  AND  HYPERBOLA  [12a 

37.  If  ^  is  the  point  on  the  auxiliary  circle  corresponding  to  the  point  P  on 
the  ellipse,  show  that  the  perpendicular  distances  of  the  foci  F^  F'  from  the 
tangent  at  Q  are  equal  to  FP  and  F'P  respectively. 

38.  If  P  is  a  point  on  the  director  circle  of  an  ellipse,  and  0  the  centre,  the 
product  of  the  distances  of  0  and  P  from  the  polar  of  P  with  respect  to  the 
ellipse  is  constant. 

39.  Show  that  the  ellipse  is  concave  towards  both  axes,  while  the  hyperbola 
is  concave  only  towards  its  transverse  axis. 

40.  Chords  are  drawn  through  the  end  of  an  axis  of  an  ellipse.  Find  the 
locus  of  their  middle  points. 

41.  If  the  eccentric  angles  of  two  points  P,  Q  on  an  ellipse  are  0i,  02,  prove 
that  the  area  of  the  parallelogram  formed  by  tangents  at  the  ends  of  diameters 
through  P  and  Q  is 

4a6csc(0i  — 02); 

and  hence  show  that  this  area  is  least  when  P  and  Q  are  the  ends  of  conjugate 
diameters. 

42.  The  sides  of  a  parallelogram  circumscribing  an  ellipse  are  parallel  to  con- 
jugate diameters  ;  prove  that  the  product  of  the  perpendiculars  from  two  opposite 
vertices  on  any  tangent  is  equal  to  the  product  of  those  from  the  other  two  vertices. 

43.  The  radius  of  a  circle  which  touches  a  hyperbola  and  its  asymptotes  ia 
equal  to  that  part  of  the  latus  rectum  intercepted  between  the  curve  and  the 
asymptotes. 

44.  Show  that  the  area  of  a  triangle  inscribed  in  an  ellipse  is 

\  a6[sin  («  —  /3)  +  sin  (jS  —  7)  +  sin  (7  —  a) ] , 

where  a,  /3,  7  are  the  eccentric  angles  of  the  vertices. 

Prove  also  that  its  area  is  to  the  area  of  the  triangle  formed  by  the  corre- 
sponding points  on  the  auxiliary  circle  as  6  :  a  ;  and  hence  its  area  is  a  maximum 
when  the  latter  is  equilateral ;  i.e.  when 

a~)3  =  i3~7  =  7'^a=f7r. 

45.  If  a  tangent  drawn  at  any  point  P  of  the  inner  of  two  similar  coaxia- 
conics  meets  the  outer  in  the  points  T  and  T\  then  any  chord  of  the  inner 
through  P  is  half  the  algebraic  sum  of  the  parallel  chords  of  the  outer  through 
rand  r. 

46.  Def.  The  two  chords  of  a  central  conic  which  join  any  point  on  the  curve 
to  the  extremities  of  any  diameter  are  called  Supplemental  Chords. 

Show  that  two  supplemental  chords  are  parallel  to  a  pair  of  conjugate 
diameters. 


CHAPTER   XI 

GENERAL  EQUATION  OF  THE  SECOND  DEGREE 

124.  It  has  been  shown  in  §  87  that  the  most  general  equation  of 
a  conic  is  the  complete  equation  of  the  second  degree.  We  shall 
now  show  that  the  general  equation, 

aar^  +  2  /io;?/  4-  &2/'  +  2  ^a;  +  2  /?/  +  c  =  0,  (1) 

can  always  be  changed  into  one  of  the  standard  forms  [§§  88-90], 
and  will  thus  prove  that  its  locus  is  always  a  conic,  eitl^er  in  one  of 
the  common  forms  or  in  one  of  the  limiting  cases.  In  order  to  do 
this  we  will  first  remove  the  terms  of  the  first  degree. 

The  equation  referred  to  parallel  axes  through  the  point  (a;',  ?/') 
will  be  found  by  substituting  x-\-x'  for  x  and  y-\-y'  for  y  [§  53],  and 
will  therefore  be,  after  collecting  terms, 

aar^  +  2  hxy  -^bf-\-2  x(ax^  ^  hy' -\- g) -[- 2  y(hx'  +  by'  +/) 

+  ax"  +  2  hxy  +  by''  +  2gx'-\-2  fy'  +  c  =  0.         (2> 
If,  as  is  generally  possible,  x'  and  y'  be  so  chosen  that 

ax'-\-hy'-{-g  =  0,  (3) 

and  hx'  +  by'-\-f=0,  (4) 

the  coefficients  of  x  and  y  in  (2)  will  both  vanish,  and  the  equation 
referred  to  (a;',  y')  as  origin  will  then  be 

aa^  +  2  hxy  +  by"^  +  c'  =  0,  (6) 

where  c'  =  ax'^  +  2  hx'y'  +  6y '2  ^  2  gx'  +  2 /?/'  +  c.  (6) 

The  locus  represented  by  (5)  is  symmetrical  with  respect  to  the 
origin  [§  28,  (9)] ;  i.e.  the  origin  is  now  at  the  centre. 

Hence  the  coordinates  of  the  centre  of  the  conic  represented  by 
(1)  are  the  values  of  x'  and  y'  which  satisfy  equations  (3)  and  (4), 

ab  -h^  ab-  h^ 

Hence,  if  h'-^ab^  0,  the  coordinates  of  the  centre  are  both  finite^ 
and  this  transformation  is  possible. 

179 


180      GENERAL  EQUATION  OF   THE   SECOND  DEGREE      [125 

Multiply  equations  (3)  and  (4)  by  a;'  and  y\  respectively,  and  sub- 
tract the  sum  from  the  right  member  of  (6) ;  we  thus  get 

c'  =  gic'  +fy'  +c.  (8) 

where  A  =  dbc  +  2  fgh  -  af^  -  hg'^  -  ch^,  (10) 

If  A  =  0,  then  c'  =  0,  and  equation  (5)  may  be  written 

ax  +  hy=  Ty^fi^  —  ab.  (11) 

Hence  the  locus  is  two  straight  lines,  which  will  be  real,  coinci- 
dent, or  imaginary  according  as  ^^  —  a&  >,  =,  or  <  0. 

If  A  =  0,  and  also  ah  —  h?  =  0,  then  c'  is  not  necessarily  zero. 

The  first  three  terms  of  equation  (5)  are  then  a  perfect  square. 
The  equation  may  therefore  be  written 

Vaic -f  Vfty  ±  V^^  =  0,  (12) 

and  represents  two  parallel  lines,  which  coincide  when  c'  =  0. 

The  function  of  the  coefiicients  denoted  above  by  the  symbol  A  is 
called  the  Discriminant  of  the  General  Equation. 

Hence  an  equation  of  the  second  degree  will  represent  two  straight 
lines  if  its  discriminant  vanishes. 

125.    Whenh^-ah^O. 

In  order  to  reduce  the  equation  [(5),  §  124] 

ax^  +  21ixy  +  hy^  +  c'  =  0,  (1) 

to  any  one  of  the  standard  forms  (§§  89,  90)  we  must  remove  the 
term  2  hxy.  For  this  purpose  we  turn  the  axes  through  a  certain 
angle  6,  keeping  the  origin  fixed. 

To  turn  the  axes  through  an  angle  6  we  substitute  for  x  and  y, 
respectively  [§  54,  (1)], 

«  cos  d  —  2/  sin  6    and    XB>mB  +  y  cos  B.  (2) 

Substituting  these  values  in  (1),  expanding  and  collecting  terms, 
we  have 


125]      GENERAL  EQUATION  OF  THE   SECOND  DEGREE      181 

(a  cos^  ^  H- 2  ^  sin  ^  cos  ^ +  &  sin^  ^)ar' 

+  2[(6  -  a)  sin  ^  cos  ^  +  ^(cos^  6  -  sin^  e)'\xy 

+  (a  sin^ ^  -  2-^  sin  d  cos  ^  +  6  cos^  e)y^  +  c'  =  0.        (3) 

The  coefficient  of  xy  in  equation  (3)  will  vanish  if  B  be  so  chosen 

*  ^*  2(6-a)sindcosd  +  2A(cos2^-sin2d)  =  0.  (4) 

i.e.  if  (a-6)sin2^  =  2^cos2^.  (5) 

.-.  tan2e=-^.  (6) 

a-b  ^  ^ 

Whence  sin  2  ^  =  ±  ^^      — ,  (7) 

V(a-6)2  4-4^2  ^ 

and  cos  2  ^  =  ±  ^~^ (8) 

Any  value  of  0  obtained  from  (6)  will  reduce  (3)  to  the  form 

a'aj2  +  6'2/2  +  c'  =  0,  or-^  +  -^  =  1,  (9) 
a'         &' 

where                a'  =  a  cos^  ^  +  2  ^  sin  ^  cos  ^  +  &  sin^  6,  (10) 

and                    6'  =  a  sin*  d  -  2  fe  sin  ^  cos  ^  +  &  cos*  0.  (11) 

Equation  (9)  is  therefore  the  required  result. 
The  values  of  a'  and  6'  may  be  expressed  in  terms  of  a,  6,  and  h 
as  follows : 
From  (10)  and  (11),  by  addition  and  subtraction,  we  obtain 

a'  +  6'  =  a4-&,  (12) 

and  a'-6'  =  (a-6)cos2^+2Asin2^.  (13) 

Substituting  (7)  and  (8)  in  (13)  gives 

2h 


a'-6'  =  ±V(a-6)*  +  4/i2  =  ^^=^.  (14) 

sin2d 

Whence,  from  (12)  and  (14), 

a'  =  ||a  +  6±V(a-6)2  +  4^|,  (15) 

and.  6'  =  ||a  +  6q:V(a-6)2  +  4/i2}.  (16) 


182      GENERAL  EQUATION  OF  THE  SECOND  DEGREE      [125 

The  ambiguity  in  the  values  of  a'  and  b'  given  by  (15)  and  (16) 
may  be  removed  by  (14).  From  the  many  values  of  $  which  satisfy 
(6)  we  will  agree  always  to  choose  that  one  which  lies  between  0** 
and  180°.  Then  0  will  always  be  an  a^ute  angle,  and  sin  2^  will 
always  he  positive.  Therefore  it  follows  from  (14)  that  a'—b'  will 
always  have  the  same  sign  as  h. 

It  is  also  worthy  of  notice  that  the  values  of  a'  and  b'  given  by 
(15)  and  (16)  are  the  two  roots  of  the  equation 

x'-(a-^b)x-  Qv"  -  ab)  =  0.  (17) 

Hence  a'  and  b'  will  have  the  same  sign  or  opposite  signs,  i.e.  the 
conic  will  be  an  ellipse  or  a  hyperbola  according  as 
A^  — a&<,  or>0. 

If  a  4-  6  =  0,  then  a'  =  —  b'  and  the  conic  is  a  rectangular  hyperbola. 

Ex.     transform  the  equation 

S x^  -\-  4:xy  +  5 y^  +  S X  -  16 y  -  IQ  =  0 
to  the  standard  form,  and  construct  the  conic. 


\^^       \     It 

>  O        /       X 


The  equations  for  finding  the  centre  are4a;  +  y  +  2  =  0  and  2 x  +  5  y  =  8. 

.-.  «'  =  -!,  y'  =  2. 
Then  c'  —  gx^  +fy'  -\- c  =  -  36. 

Therefore  the  equation  referred  to  parallel  axes  O'JP,  O'T  through  the 


126]      GENERAL  EQUATION  OF  THE  SECOND  DEGREE      183 

centre  is  8x^  +  ixy -{■  5y'^  =  Z6. 


Also        a'  =  ^\a  +  b±  V(a -6)2  +  d/i^  j  =  J  (13  ±  5)  =  9  or  4, 

6'  =  I  {a  +  6  T  V(a-6)2  +  4^2J  =  J(13  T  6)  =  4  or  9. 

Since  h  is  positive,  we  take  a'  =  9  and  6'  =  4. 

Hence  the  equation  of  the  curve  referred  to  its  own  axes  O'X",  0'  T'  as  axes 

of  coordinates  is  «     ,,„ 

-  +  ^  =  1. 
4^9* 

Also  tan2^=»^^=i 

a-b     3 

Therefore  the  line  O'X"  must  be  drawn  so  that  Z  X'O'X"  =  ^  tan-i  |, 

126.   Whenh^-ab  =  0. 

In  this  case  the  coordinates  of  the  centre  [(7),  §  124]  are  both 
infinite,  and  therefore  the  first  degree  terms  cannot  be  removed  by 
changing  to  a  new  system  of  axes  parallel  to  the  old. 

Since  the  second  degree  terms  now  form  a  perfect  square,  the 
general  equation  may  be  written 

(l3y-{-axy  +  2gx  +  2fy  +  c  =  0,  (1) 

where  a  =  -y/ay  p=^b,  a  has  the  same  sign  as  h,  and  fi  is  always 

P°«^^i^®-  .-.  h  =  a/3.  (2) 

First  Method.    From  equation  (6),  §  125,  we  have 

tan2fl-   ^^'  __2«^_^tan^  .„. 

tan2^---^^-^-^^_^— ^-^.  (3) 

.-.  taii<?  =  |,or-^.  (4) 

If  we  turn  the  axes  through  an  angle  given  by  either  of  these 
values  of  tan  d,  the  coefficient  of  icy  in  the  new  equation  will  vanish. 

If  we  take  ^  =  tan~^[  —  —  j,  the  equation  of  the  new  ic-axis  will  be 

ax-{-/3y  =  0.  (5) 

We  will  use  this  value,  and  will  agree  always  to  take  the  positive 
direction  of  the  new  a^axis  so  that  $  shall  be  numerically  less  than 
90°.     Then  $  will  be  positive  or  negative  according  as  h  (or  a)  is 


184      GENERAL  EQUATION  OF  THE  SECOND  DEGREE      [126 
negative  or  positive,  and  we  have  from  (4) 
sin  6  = ,       cos  ^  = 


Hence,  to  turn  the  axes  through  an  angle  $  thus  chosen,  we  must 
substitute  for  x  and  y,  respectively  [§  54,  (1)], 


■Va^  +  P'  -y/a'  +  P' 

Substituting  the  expressions  (6)  in  (1)  gives 

(a'  +  ^f  +  2-^f^x  +  2^^i^y  +  c  =  0.  (7) 

Va^  +  jS  Wa^  +  p^ 

Completing  the  square  in  the  terms  containing  y,  equation  (7)  may 
be  reduced  to  the  form 

where  ^^c(.^  +  ^y- («,  +  ff/)^ 

and  ^^_     «^  +  ^/ 

If  now  the  origin  be  moved  to  the  point  {H,  K),  equation  (8)  will 
take  the  standard  form 

y2  =  2  f-^=x.  (9) 

Therefore  equation  (1)  represents  a  parabola  whose  axis  is  par- 
allel to  the  line  (5),  and  whose  latus  rectum  is 

Second  Method.    Equation  (1)  may  be  written 

(ax  +  l3y-{-\y  =  2(a\-g)x  +  2(l3\-f)y  +  \'--c,         (10) 

where  X  is  any  constant,  for  which  a  particular  value  will  now  be 
determined. 
We  observe  that  the  line  whose  equation  is 


126]      GENERAL  EQUATION  OF  THE  SECOND  DEGREE      185 

ax  +  py-{-k  =  0  (11) 

is  parallel  to  the  axis  of  the  parabola  [see  (5)  above]  for  all  values 
of  A.    Hence  we  will  choose  A.  so  that  the  straight  line 

2(ak-9)x-\-2{p\-f)y  +  X'-c  =  0  -     ^12) 

shall  be  perpendicular  to  the  line  (11). 
The  lines  (11)  and  (12)  wiU  be  at  right  angles  (§  45)  if 

-if  ^  =  ^-A.  (13) 

With  this  value,  Xi,  equation  (10)  may  be  written 

{ax  +  py  +  X,y  =  2^:z^(fix-ay  +  K),  (14) 

where  K^^±^(bl^\  (ig) 

Changing  the  linear  expressions  in  (14)  to  the  distance  form  gives 
/ax±Ji^y  ^        af--pg     /px-ay  +  K\  .^q. 

If  now  we  take  the  lines 

ax  +  py  +  Xj^  =.0  (17) 

and  px-ay  +  K=0  (18) 

for  new  axes  of  x  and  y,  respectively,  the  new  equation  will  be 

y2  =  2     «/-Pg^a..  (19) 

Hence  (17)  represents  the  axis  of  the  parabola,  and  (18)  the  tan- 
gent at  the  vertex.  The  curve  will  lie  on  the  positive  or  negative  side 
of  the  line  (18)  according  as  (af—Pg)  is  positive  or  negative. 

If,  a\i  —  g  =  p\i  —  g=  0,  the  line  (12)  cannot  be  determined. 
But  in  this  case  equation  (10)  reduces  to 

(ax  +  py  +  k,y  =  Xr*-c,  (20) 

that  is,  the  conic  then  consists  of  two  parallel  lines. 


186      GENERAL  EQUATION  OF  THE  SECOND  DEGREE       [126 

Ex.     Find  the  standard  form  of  the  equation 

(4  y  -  3  x)2  -  20  a;  +  110  y  -  75  =  0.  (1) 


First  Method.    Take  4  y— 3a;  =  0  as  the  new  jc-axis;  i.e.  turn  the  axes 
through  an  angle  0,  such  that  tan  ^  =  |,  and  therefore  sin  tf  =  |,  cos  ^  =  |. 
Then  the  formulae  of  transformation  are 

4x'-Sy[ 


x  =  x'  cos  e  —  y'  sme  = 


and 


y  =  aj'  sin  ^  +  y'  cos  6 


Sx'  +  4y' 
5 
Substituting  these  values  in  equation  (1),  it  becomes 


y'2  +  2  5c'  +  4 


3  =  0, 


(2) 


or  (y'  +  2)2  =  -  2(x'  - 1),  , 

which  is  the  equation  of  the  curve  referred  to  the  new  axes  OX,  Or'. 

Moving  the  origin  to  the  point  O'  (|,  —  2),  with  respect  to  the  new  axes,  we 
obtain  from  (2)  the  required  equation 

y"2  =  -  2  x". 
Hence  the  curve  is  on  the  negative  side  of  the  y-axis  0'  T". 
Second  Method.    The  given  equation  (1)  may  be  written 

(4  y  -  3  a;  +  X)2  =  (20  -  6  X)x  +  (8  X  -  110)y  +  X^  +  75. 
We  will  now  determine  X  so  that  the  two  lines 
4y-3a:  +  X  =  0 
and  (20  -  6  X)a;  +  (8  X  -  110)y  +  X^  +  75  =  0 

shall  be  at  right  angles. 


(3) 


(4) 

(5) 
(6) 


126]      GENERAL  EQUATION  OF  THE  SECOND  DEGREE      187 

The  required  value  of  X  is  given^by  the  equation  (§  45) 
-  3(20  -  6  X)  +  4  (8  X  -  110)  =  0. 
.-.   X  =  10. 
With  this  value  of  X  equation  (4)  becomes 

(4y  _3a;-f  10)2  =  - 10(4  x  +  3y-17i), 
^,  /iv-^+ioy ^ _  2  fix  +  3y-n\\  (7^ 

Draw  the  lines 

4y-3a;  +  10  =  0,  O'X",  (8) 

and  4a;  + 3  y- 17i  =  0,  O'F'.  (9) 

These  lines  are  at  right  angles.    If  we  take  (8)  as  the  new  x-azis  and  (9)  as 
the  new  y-axis,  the  equation  of  the  curve  will  be 

y2  =  -  2  x.  (10) 

Therefore  the  locus  is  a  parabola  whose  latus  rectum  is  2,  and  lies  on  the 
negative  side  of  the  line  (9). 

EXAMPLES 

Construct  the  following  conies  by  transforming  the  equations  to  their  standard 
forms : 

1.  (4y-3«)«  +  4(4a;  +  3y)  =  0.  3.   3a;a  +  2a:y +  3ya  =  8. 

2.  (3a;-4y-12)2  =  16(4a;  +  3y).  4.   x^-6xy  +  y^  =  lQ. 

6.  4a;«-24a;y  +  lly2-16a;-2y-89  =  0.  7.  2 x^ -\- ixy -\- 6 y^  =  3Q. 

,    6.  5a;2-4xy  +  8y2-22iB  +  16y-10  =  0.  8.  8x2-6a;y-4  y2  =  34. 

9.  9a;«-12xy  +  4ya  =  10(2x  +  3y  +  5). 

10.  3a;2-2xy  +  2y2-16x-8y  +  8  =  0. 

11.  6x«  +  24xy-y2  +  50y-56  =  0.  16.  2x2  +  a^  +  3y2  =  23. 

12.  a;a-2xy  +  y2-5x-y-2  =  0.  16.  xy +  3x-6y +  5  =  0. 

18.  x2-6xy  +  9y2-2x  +  6y  +  l=0.  17.   x2  +  4y2  +  4x  =  0. 
14.  4x2  +  4xy  +  y2-f4x-3y  +  4  =  0.  18.  4x3-9y2  +  24x  =  0. 

19.  24xy  +  7y2-6(8x-10y-9)=0. 

20.  25x«-20xy  +  4y2  +  5x-2y-6  =  0. 

21.  2x!»  +  7xy-4y24-4x  +  7y-18J  =  0. 

22.  2x2  +  xy-6y2-5x+lly-3  =  0.  26.   xa-2xy-y2  =  20. 
28.  x2  +  2xy  +  ya-'12x  +  2y-3  =  0.  26.    (5 y  +  12 x)^  =  102 x. 
M.  a:a-xy-2ya-x-4y-2  =  0.  27.   x2-4x-3y  =  5. 


188  MISCELLANEOUS  EXERCISES  [126 


EXAMPLES  ON   LOCI 

1.  Show  that  the  locus  of  a  point,  the  sum  of  the  squares  of  whose  distances 
from  n  fixed  points  is  constant,  is  a  circle. 

2.  Find  the  locus  of  the  centre  of  a  variable  circle  which  touches  a  fixed 
circle  and  a  fixed  straight  line. 

3.  Find  the  locus  of  the  centre  of  a  circle  which  touches  two  fixed  circles. 
Four  cases  should  be  considered.  What  does  the  locus  become  when  the  fixed 
circles  are  equal  ? 

4.  Find  the  locus  of  the  middle  points  of  all  chords  of  a  given  circle  which 
pass  through  a  fixed  point.     [Take  the  fiLxed  point  as  pole.] 

6.  A  straight  rod  moves  so  that  its  ends  constantly  touch  two  fixed  perpen- 
dicular rods.     Find  the  locus  of  any  point  P  on  the  moving  rod. 

6.  On  a  level  plain  the  crack  of  a  rifle  and  the  thud  of  the  ball  striking 
the  target  are  heard  at  the  same  instant.  Find  the  locus  of  the  hearer. 
[S.  L.  Loney's  Coordinate  Geometry,  p.  283.] 

7.  In  a  given  circle  let  AOB  be  a  fixed  diameter,  OC  B.nj  radius,  CD  the 
perpendicular  from  C  on  AB  ;  let  P  and  Q  be  two  points  on  the  line  through  O 
and  C  such  that  QO  =  OP  =  DC.  Find  the  locus  of  P  and  Q  as  OC  turns 
about  O. 

8.  A  and  B  are  two  fixed  points,  and  P  moves  so  that  PA  =  n  •  PB.  Find 
the  locus  of  P. 

9.  AOB  and  COD  are  two  straight  lines  which  bisect  one  another  at  right 
angles.    Find  the  locus  of  a  point  P  such  that  PA-  PB  =  PC  •  PD. 

10.  If  ABC  is  an  equilateral  triangle,  find  the  locus  of  a  point  P  such  that 
P^2  =  p^_|.pce. 

11.  AB  is  a  fixed  diameter  of  a  given  circle  and  -<4.C  is  any  chord  ;  P  and  Q 
are  two  points  on  the  line  AG  such  that  QG  =  CP  =  CB.  Find  the  locus  of  P 
and  Q2&  AC  turns  about  A. 

12.  Any  straight  line  is  drawn  from  a  fixed  point  0  meeting  a  fixed  straight 
line  in  P,  and  a  point  Q  is  taken  in  this  line  such  that  OP  •  OQ  is  constant. 
Find  the  locus  of  Q. 

13.  Any  straight  line  is  drawn  from  a  fixed  point  O  meeting  a  fixed  circle  in 
P,  and  on  this  line  a  point  Q  is  taken  such  that  OP  •  0§  is  constant.  Show 
that  the  locus  of  Q  is  a  circle.     [See  suggestion  under  Ex.  4.] 

14.  Find  the  locus  of  a  point  such  that  the  sum  of  the  squares  of  its  distances 
from  the  sides  of  an  equilateral  triangle  is  constant. 

15.  The  square  of  the  distance  of  a  point  P  from  the  base  of  an  isosceles 
triangle  is  equal  to  the  product  of  its  distances  from  the  other  two  sides.  Find 
the  locus  of  P. 


128]  MISCELLANEOUS  EXERCISES  189 

MISCELLANEOUS  PROBLEMS  ON  LOCI 

1.  Show  that  the  curve  on  the  concave  side  of  the  new  moon  is  an  ellipse. 

2.  A  circular  cylinder  rolls  along  on  a  plane  surface.  Find  the  locus  of  the 
point  of  contact  between  the  plane  surface  and  an  oblique  plane  section  of  the 
cylinder. 

3.  What  kind  of  a  curve  must  be  used  in  making  a  pattern  for  cutting  elbows 
of  stovepipe  from  sheet  iron  ? 

4;  If  the  hub  of  a  cart-wheel  is  not  perpendicular  to  the  plane  of  the  wheel, 
what  kind  of  a  curve  is  the  track  of  the  wheel  on  a  level  road  ?  Is  this  problem 
the  same  as  No.  2  ?    If  not,  why  ? 

5.  If  a  wheel  is  rotating  on  a  fixed  axle  with  a  uniform  angular  velocity, 
while  a  fly  is  crawling  outward  along  a  spoke  of  the  wheel  at  a  constant  rate, 
what  is  the  equation  of  the  locus  of  the  fly  in  the  plane  ? 

6.  If  a  spiral  spring  rolls  on  a  plane  surface,  what  kind  of  a  track  does 
it  make  ? 

7.  What  kind  of  a  curve  is  the  shadow  of  a  spiral  spring,  if  the  rays  of  light 
are  all  perpendicular  to  the  plane  of  the  shadow,  and  the  axis  of  the  spring  parallel 
to  the  plane  ? 

8.  The  curve  described  by  a  piece  of  paper  sticking  to  the  rim  of  a  cart- 
wheel as  the  wheel  rolls  along  in  a  straight  line  on  a  level  road  is  called  a 
Cycloid. 

Take  the  origin  at  the  pomt  where  the  piece  of  paper  was  originally  on  the 
ground,  and  use  the  wheel's  track  as  the  x-axis  ;  let  6  represent  the  angle  through 
which  the  wheel  has  turned,  and  a  the  radius  of  the  wheel.    Then  show  that 

X  =  a{d  —  sin  ^),  and  y  =  a(l  —  cos  6). 

Eliminate  0  and  show  that  the  equation  of  the  cycloid  is 


x  =  a  vers-i  -  —  V2  ay  —  y^. 


9.  In  2  minutes  after  leaving  a  station  a  railroad  train  attains  a  speed  of 
40  miles  an  hour,  which  it  maintains  for  3  minutes ;  then  it  strikes  a  grade,  and 
in  1  minute  its  speed  is  reduced  to  30  miles,  which  it  maintains  for  3  minutes ; 
in  1  minute  more  it  slows  down  and  stops  at  the  next  station.  Draw  a  curve 
whose  ordinate  shall  represent  approximately  the  speed  of  the  train.  What  is 
the  approximate  distance  between  the  stations  ? 

10.  In  a  steel  bar  the  stretch  varies  as  the  strain  until  the  elastic  limit  is 
reached.  From  this  on  the  stretch  varies  at  a  greater  and  greater  rate  with 
regard  to  strain  until  the  bar  snaps.    Draw  a  curve  which  will  illustrate  this  law. 


190  MISCELLANEOUS  EXERCISES  [126 

11.  The  specific  heat  of  ice  is  .5,  of  water  1.  It  requires  80  calories  of  heat  to 
convert  1  gram  of  ice  at  0°  C.  into  water  at  0°  C,  and  about  600  calories  to  con- 
vert 1  gram  of  water  at  100°  C.  into  steam  at  100°  C.  Draw  a  curve  whose 
ordinate  shall  show  the  change  in  temperature,  per  calorie^  at  every  stage  of  the 
process  as  1  gram  of  ice  at  —  40°  C.  is  converted  into  steam. 

12.  It  is  a  law  of  physics  that  the  product  of  the  volume  and  pressure  of  a 
gas  is  constant.  Construct  the  graphical  representation  of  this  law.  What  kind 
of  a  curve  is  it  ? 

13.  Suppose  the  steam  is  allowed  to  enter  a  cylinder  during  only  one-fourth 
of  the  stroke.  Draw  a  curve  whose  ordinate  shall  represent  theoretically  the 
pressure  on  the  piston. 

14.  Waves  from  two  different  centres  have  the  same  length.  Find  the  locus 
of  the  points  where  crest  coincides  with  crest  and  where  trough  meets  trough. 
Find  also  the  locus  of  the  points  where  the  crests  of  one  wave  coincide  with  the 
troughs  of  the  other. 

16.  Find  the  locus  of  all  points  in  a  plane  that  are  equally  illuminated  by  two 
lights  situated  in  the  plane.     What  is  the  locus  in  space  ? 

16.  If  a  vertical  tube  is  moved  horizontally  with  a  uniform  velocity,  while  a 
ball  is  falling  freely  through  the  tube,  what  is  the  path  described  by  the  ball  ? 
(We  here  assume  that  the  student  is  familiar  with  the  law  of  falling  bodies, 
viz.  s  =  Jsr«2.) 

17.  Show  that  the  equation  of  the  path  of  a  projectile  fired  from  a  gun  with 
an  initial  velocity  of  c  feet  per  second,  6  being  the  angle  of  elevation  of  the  gun,  is 

y  =  a;tane-^sec2^, 

the  origin  being  at  the  muzzle  of  the  gun,  and  the  a^axis  horizontal. 

Find  also  the  horizontal  range  of  the  projectile,  and  show  that  any  two  com- 
plementary angles  of  elevation  will  give  the  same  range. 
Show  also  that  the  range  is  a  maximum  when  6  =  45°. 

Show  that  the  angle  of  elevation  required  to  strike  a  given  point  («',  y')  is 
given  by  the  formula 

tan  g  ^  c2  j,  Vc^  -  g^'^  -  2  c^gy' 
gx' 

18.  Draw  a  curve  which  will  show  the  relation  between  a  man's  daily  wages 
and  the  number  of  days  he  must  work  in  order  to  be  able  to  meet  his  necessary 
annual  expenses. 

19.  The  manager  of  a  gas  plant  finds  that  his  customers  will  spend  annually 
a  fixed  sum  for  gas.  Find  the  curve  that  will  show  the  relation  between  the 
price  of  gas  and  the  quantity  of  gas  that  can  be  sold. 


126]  MISCELLANEOUS  EXERCISES  191 

20.  The  annual  expense  of  a  business  firm  for  rent,  interest,  taxes,  insurance, 
depreciation  of  plant,  etc.,  is  practically  constant.  Draw  a  curve  which  will 
show  the  relation  between  the  daily  volume  of  business  and  the  number  of  days 
necessary  to  run  the  business  in  order  to  meet  these  fixed  charges. 

21.  The  revenue  from  the  sale  of  a  commodity  is  the  product  of  the  price  by 
the  quantity  sold.  Using  price  as  ordinate,  draw  a  curve  which  shall  show  the 
relation  between  the  price  (p)  and  the  quantity  sold  (5),  if  the  revenue  (i?)  is 
constant. 

Such  a  curve  may  be  called  a  Constant  Bevenue  Curve.  What  kind  of  a 
curve  is  it  ? 

22.  Draw  a  curve  showing  the  relation  between  the  demand  for  a  commodity 
and  its  market  price,  using  price  for  ordinate.  (Demand  Curve.)  Draw  another 
curve  showing  the  relation  between  the  supply  and  the  cost  of  production  of  this 
same  commodity,  using  cost  for  ordinate.  (Supply  Curve.)  What  is  indicated 
by  the  point  of  intei-section  of  these  two  curves  ?  Draw  a  third  curve  whose 
ordinate  shall  be  the  ordinate  of  the  demand  curve  minus  the  ordinate  of  the 
supply  curve.  (Monopoly  Bevenue  Curve.)  What  does  the  ordinate  of  this 
third  curve  represent  ?  Now  construct  a  constant  revenue  curve  (see  Ex.  21) 
which  shall  touch  this  last  curve.  What  is  the  significance  of  this  point  of 
contact?  (See  Principles  of  Economics  by  Alfred  Marshall,  Vol.  I,  3d  Ed., 
p.  636.) 

23.  Find  the  equation  of  a  curve  whose  ordinate  shall  represent  the  amount 
of  a  given  principal  at  a  fixed  rate  of  compound  interest,  using  time  as  abscissa. 


SOLID  GEOMETRY 


CHAPTER  XII 

SYSTEMS  OF  COORDINATES,   THE  POINT,   RECTANGULAR 
COORDINATES 


127.  In  the  rectangular  system  of  coordinates,  three  mutually 
perpendicular  planes  XOT,  TOZy  ZOX  are  chosen  as  planes  of 
reference.  These  planes  are  called  Coordinate  Planes ;  their  lines  of 
intersection  OX,  0  F,  OZ,  Coordinate  Axes ;  and  their  point  of  inter- 
section, 0,  the  Origin. 


The  position  of  a  point  P  in  space  is  then  completely  determined 
when  its  distances  APy  BP,  CP,  from  each  of  these  planes,  measured 
parallel  to  the  coordinate  axes,  and  the  direction  in  which  these 
distances  are  measured,  are  given.  These  three  lines,  or  the  num- 
bers which  represent  them,  are  called  the  Rectangular  Coordinates  of 
the  point  P,  and  are  always  written  in  the  order  (x,  y,  z). 

193 


194  COORDINATES  [127 

We  shall  consider  distances  positive  when  measured  in  the  direc- 
tions OX,  OT,  or  OZ;  that  is,  to  the  right,  forward,  or  upward. 
Then  distances  measured  in  the  opposite  directions  will  be  negative. 

The  coordinate  planes  divide  all  space  into  eight  equal  compart- 
ments, which  may,  for  convenience,  be  called  Octants.  The  octant 
0-XYZ  is  called  the  first,  but  there  is  no  established  order  for 
numbering  the  others. 

The  position  of  any  point  P(a,  6,  c)  (a,  6,  and  c  being  positive 
numbers)  may  be  found  as  follows:  measure  on  the  axes  the  dis- 
tances OD  =  a,  OE  —  hy  OF=c,  and  through  the  points  D,  E,  F 
draw  planes  parallel  to  the  coordinate  planes,  forming  a  rectangular 
parallelopiped ;  the  intersection  of  these  three  planes  will  be  the 
required  point  P. 

There  are  seven  other  points  whose  absolute  distances  from  the 
coordinate  planes  are  the  same  as  those  of  P.  What  are  their  coor- 
dinates ?  What  do  these  eight  points  form  ?  What  are  the  coor- 
dinates of  the  points  A,  B,  C,  D,  E,  F? 

Moreover,  it  is  obvious  that  x  =  a  for  all  points  in  the  plane 
PBDO  indefinitely  extended;  also  that  x  =  a  and  y  =  b  for  all 
points  on  the  indefinite  line  PC.  Or,  in  other  words,  a;  =  a  is  the 
equation  of  the  plane;  x  =  a,  y  —  h  are  the  equations  of  the  line 
PC'y  while  x  =  a,  y  =  b,  z  =  c  are  the  equations  of  the  point  P. 
Thus,  the  more  the  location  of  a  point  is  restricted,  the  greater  the 
number  of  equations  its  coordinates  must  satisfy.  What  are  the 
equations  of  the  other  faces  of  this  parallelopiped?  the  other  edges? 

It  is  easy  to  see  that  the  system  of  rectangular  coordinates  in  a 
plane  is  a  special  case  of  the  more  general  system  here  described, 
in  which  one  of  the  coordinates  has  become  zero.  Hence  we  shall 
find  that  we  can  reduce  formulae  in  solid  geometry  to  the  cor- 
responding formulae  in  plane  geometry  by  placing  z  equal  to  zero. 
The  student  should  bear  this  constantly  in  mind. 

EXAMPLES 

1.  What  are  the  equations  of  the  coordinate  planes  ?  the  coordinate  axes  ? 

2.  What  is  the  locus  of  the  point  (x,yjz)U.x  =  y?  y  =  z?  z  =  x?  x  =  -y? 
yzz  —  z?  z  =-  xf 


129] 


COORDINATES 


195 


8.  What  is  the  locus  of  the  point  {x^  y,  z)  if  x  =  y  =  2!?  x  =  —  y  =  «? 
x=y=-z?  x=-y=-z? 

Let  P  in  the  figure  be  the  point  (a,  b,  c). 

4.  Show  that  for  every  point  in  OP  -  =  ^  =  -. 

a     0     c 

6.   Show  that  the  equation  of  the  plane  ABDE  is-  +  ^  =  l.  ~~ 

a     b 

6.   Find  the  equations  of  the  planes  OFPd  ODPA,  and  OEPB. 

128.  To  find  the  coordinates  of  a  point  which' divides  the  straight 
line  joining  two  given  points  in  a  given  ratio  7% :  ini^ 

Let  (xif  ^1,  z^  and  {x^,  ^2?  ^2)  be  the  two  given  points,  and  (a;,  y,  z) 
the  required  point. 

The  proof  is  precisely  the  same  as  that  given  for  the  correspond- 
ing theorem  in  plane  geometry  (§  9).     The  results  are 


05  = : 5    y  - 


mi  +  m^  mi  +  wt2  ^wi  +  w^ 

If  (x,  y,  z)  is  the  middle  point  of  the  line,  then 


a,=^i+£?,   j,=l?i  + 


»  = 


(2) 


129.  To  find  the  distance  between 
two  points  whose  rectangular  co- 
ordinates are  given. 

Let  Pi(a;„  2/1,  2i)  and  PgCa^  ^2,  ^^ 
be  the  given  points. 

Through  the  points  Pi  and  P2  draw- 
planes  parallel  to  the  coordinate 
planes,  forming  a  rectangular  par- 
allelopiped,  whose  diagonal  is  P1P2, 
and  whose  edges  P2Q,  QB,  RPi  are 
parallel  to  the  axes. 

Then  PaPi^  =  P2Q'  4-  QR^  +  RPi^- 

But  P2Q  =  »!  —  X2,  QR  =  yi  —  Vii  and  RP^  =  z^  —  z2. 


/     '/ 

^.^ 

/■    [ 

then 


.-.  P2P1  =  y/{X\  -  a32)2  +  (1/1  -  1/2)2  +  (21  -  »2)2.  (1) 

If  p  represents  the  distance  of  any  point  (a?,  y,  2)  from  the  origin, 
P  =  Vaj2  +  2/2  +  »2.  (2) 


196  COORDINATES  [130 

EXAMPLES 

1.  Show  that  the  coordinates  of  the  centre  of  gravity  of  the  triangle  whose 
vertices  are  (xi,  yi,  z{),  (X2, 2/2,  22),  and  (xs,  ys,  zz)  are 

xi  +  a;2  +  xz   yi  +  ^2  +  yz  and  ^1  +  ^^^  +  zz 
3*3'  3 

2.  Show  that  the  four  lines  which  join  the  vertices  of  a  tetrahedron  to  the 
centres  of  gravity  of  the  opposite  faces  meet  in  a  point  which  divides  the  lines 
in  the  ratio  3  : 1.     (This  point  is  the  centre  of  gravity  of  the  tetrahedron.) 

3.  Show  that  the  centre  of  gravity  of  any  tetrahedron  bisects  each  of  the 
three  lines  joining  the  middle  points  of  the  opposite  edges.  "What  does  this 
theorem  become  if  we  consider  the  four  points  as  vertices  of  a  twisted  quadri- 
lateral ?    What  when  the  fourth  point  moves  into  the  plane  of  the  other  three  ? 

4.  Show  that  the  sum  of  the  squares  of  the  diagonals  of  any  quadrilateral  is 
twice  the  sum  of  the  squares  of  the  lines  joining  the  middle  points  of  the 
opposite  sides.     State  the  corresponding  theorem  for  a  tetrahedron. 

6.  Show  that  the  sum  of  the  squares  of  two  pairs  of  opposite  edges  of  a 
tetrahedron  is  equal  to  the  sum  of  the  squares  of  the  third  pair  of  opposite 
edges  plus  four  times  the  square  of  the  line  joining  the  middle  points  of  the 
third  pair.  What  is  the  corresponding  theorem  for  a  twisted  quadrilateral? 
For  a  plane  quadrilateral  ? 

Orthogonal  Projections 

130.  The  points  A,  J5,  C  (§  127)  are  the  projections  of  P  on  the 
three  coordinate  planes ;  while  D,  E,  F  are  its  projections  on  the 
axes.  The  projection  of  any  locus  on  a  given  plane  is  the  locus  of 
the  projections  of  all  the  points  of  the  given  locus.  The  angle 
between  a  straight  line  and  a  plane  is  the  angle  the  line  makes  with 
its  projection  on  the  plane.  Hence,  from  plane  geometry,  the  pro- 
jection of  a  limited  line  on  any  plane  is  equal  to  the  line  multiplied 
by  the  cosine  of  the  angle  between  the  line  and  the  plane. 

The  projection  of  a  limited  line  on  an  a>xis  (any  other  line)  is 
that  part  of  the  axis  intercepted  between  two  planes  through  the 
ends  of  the  line  perpendicular  to  the  axis.  The  projections  of  a 
line  on  a  series  of  parallel  axes  are  evidently  all  equal.  The  angle 
between  two  lines  which  do  not  intersect  is  equal  to  the  angle 
between  two  intersecting  lines  parallel  respectively  to  the  two  given 
lines. 


131] 


COORDINATES 


197 


Hence,  as  in  plane  geometry,  the  projection  of  a  limited  line  on  any 
axis  is  equal  to  the  line  multiplied  by  the  cosine  of  the  angle  between 
the  line  and  the  axis. 

Also,  the  projection  of  a  broken  lijie  (in  space)  on  any  axis  is  equal 
to  the  projection^  on  the  same  axiSj  of  the  straight  line  joining  the  ends 
of  the  broken  line. 

For  example,  let  p  be  the  distance  from  the  origin  to  the  point 
(Xj  y,  z).     Then,  projecting  on  any  line  we  get 

Proj.  of  p  =  Proj.  of  a?  +  Proj.  oty  +  Proj.  of  z,  (1) 

This  equation  is  evidently  true  if  p  is  the  diagonal,  and  a;,  y,  z  are 
the  three  dimensions  of  any  rectangular  parallelopiped.  We  shall 
frequently  have  occasion  to  use  this  special  case  of  the  last  theorem. 

Polar  Coordinates.    Direction  Cosines 

131.  Let  P(x,  y,  z)  be  any  point  in  space  referred  to  rectangular 
axes. 

The  position  of  P  will  evidently 
be  determined  if  we  know  its  dis- 
tance p  from  the  origin  and  the 
angles  «,  )8,  y,  which  OP  makes 
with  the  axes.  The  four  quantities 
0>>  «>  Pf  y)  a.re  the  Polar  Coordinates 
of  P.  The  distance  p  is  called  the 
Radius  Vector  of  the  point  P,  and 
a,  /8,  y  are  called  the  Direction  Angles 
of  the  line  OP. 

Since  a;,  2/,  and  z  are  the  projec- 
tions of  p  on  the  three  axes,  we  have 

05  =  pcos a,  y  =  p cos P,  »  =  p COSY,  (1) 

Cos  a,  cos  Pj  and  cos  y  are  called  the  Direction  Cosines  of  the  line 
OP.  Hereafter  we  shall  represent  them  by  the  letters  Z,  m,  and  n, 
respectively.     Then  equations  (1)  become 

aj  =  ip,  y  =  mp,  z  =  np.  (2) 

It  is  to  be  carefully  noticed  that  Z,  m,  n  are  the  direction  cosines 
of  a  directed  line;  that  if  the  signs  of  l,  m,  n  are  all  changed,  the 


198  COORDINATES  [132 

direction  of  the  line  is  reversed.  It  is  evident  from  equation  (2)  that 
the  signs  of  I,  m,  and  n  for  any  line  through  the  origin  will  be  the 
same  respectively  as  the  signs  of  the  rectangular  coordinates  x,  y, 
and  z  of  any  point  P  on  the  line,  provided  OP  be  taken  as  the  posi- 
tive direction  of  the  line.  Hence  we  may  always  choose  the  polar 
coordinates  of  a  point  so  that  p  shall  be  positive,  and  each  of  the 
angles  a,  /8,  y  shall  be  less  than  180°. 

The  direction  cosines  of  any  line  are  evidently  the  same  as  the 
direction  cosines  of  a  parallel  line  through  the  origin,  since  parallel 
lines  make  the  same  angles  with  the  axes. 

Squaring  and  adding  equations  (2)  we  get 

p\V^  +  w?-\-n^^;>?-^f  +  z'',  (3) 

and,  since  p^  =  a^  +  /  +  z\  [(2),  §  129]  we  have 

l^  +  m^  +  n^^l.  (4) 

That  is,  the  sum  of  the  squares  of  the  direction  cosines  of  any  line  is 
equal  to  unity. 

Hence  the  four  polar  coordinates  of  a  point  are  equivalent  to  only 
three  independent  conditions. 

If  we  divide  each  of  the  three  numbers  a,  6,  c  by  the  square  root 
of  the  sum  of  their  squares,  we  get 

Since  these  results  are  numbers  which  satisfy  equation  (4),  they  are 
the  direction  cosines  of  some  line,  whatever  the  values  of  a,  6,  c 
may  be. 

That  is,  any  three  numbers  are  proportional  to  the  direction  cosines 
of  some  line. 

Note.  —  Custom  is  not  uniform  in  regard  to  the  use  of  the  name  Polar 
Coordinates.    Many  authors  apply  the  name  to  the  system  described  in  §  132. 

Spherical  Coordinates 

132.  Let  OX,  OF,  OZ,  be  a  set  of  rectangular  axes,  and  P  any 
point.  Then  OP,  or  p,  the  angle  6  which  OP  makes  with  OZ,  and 
the  angle  <^  which  the  plane  ZOP  makes  with  the  fixed  plane  XOZ 
are  the  Spherical  Coordinates  of  the  point  P,  and  are  written  (p,  6,  <^). 


133] 


COORDINATES 


199 


Since  OC  =  p  sin  Oj  the  relations  between  rectangular  and  spherical 
coordinates  are 

0?  =  p  sin  6  cos  <i>,    2/  =  p  sin  6  sin  <f>, 
iS  =  pCO80.  (1) 

Whence  the  relations  between 
polar  and  spherical  coordinates 
are  found  by  equation  (1)  §  131 
to  be 

cos  (x  =  sin  6  cos  <|>,  cos  §  =  sin  0  sin  <f>, 
7  =  e.  (2) 

If  P  is  a  point  on  the  surface 
of  the  earth  and  Z  the  pole,  then  $ 
is  the  co-latitude  and  <f>  the  longi- 
tude of  P.  If  P  is  a  point  on  the 
celestial  sphere  and  Z  the  pole,  0  is  the  co-declination  and  <^  the  right 
ascension  of  P;  if  ^  is  the  zenith,  then  0  is  the  zenith  distance  and  <^ 
is  the  azimuth  of  P. 

133.  Cylindrical  Coordinates.  —  If  the  position  of  the  foot  of  the 
coordinate  z  in  the  plane  xy  is  defined  by  the  polar  coordinates  (p,  6) 
instead  of  (a?,  y),  then  (p,  0,  z)  are  called  Cylindrical  Coordinates. 

EXAMPLES 

1.  Find  the  direction  cosines  of  a  line  equally  inclined  to  the  three  axes. 

2.  A  line  makes  an  angle  of  60°  with  each  of  two  axes.  What  angle  does  it 
make  with  the  other  axis  ? 

3.  If  one  direction  angle  of  a  line  is  135°,  another  120°,  what  is  the  third  ? 

4.  What  are  the  direction  cosines  of  a  line  perpendicular  to  the  a>axis  ?  the 
y-axis  ?  the  2;-axis  ? 

5.  What  are  the  direction  cosines  of  a  line  parallel  to  the  x-axis  ?  the  ?/-axis  ? 
the  5j-axis  ? 

6.  Find  the  direction  cosines  of  the  line  joining  the  origin  to  the  point 
(3,  -  2,  -  1).    Of  the  line  joining  the  points  (-  2,  4,  2)  and  (1,  2,  -  4). 

7.  Find  the  direction  cosines  of  the  line  joining  the  two  points  (xi,  yi,  zi) 
and  (X2,  yz,  22)- 

8.  Show  that  the  square  of  the  distance  between  the  two  points  whose  polar 
coordinates  are  (pi,  «!,  Pi,  71)  and  (p2,  a2»  P2,  .'2)  is 

pi^  +  />2^  —  2  pi/)2(cos  «!  cos  a2  +  COS  ft  COS  P2  +  cos  7i  cos  72). 


CHAPTER  XIII 
LOCI 

134.  We  have  seen  in  §  127  that  x  =  ais  the  equation  of  a  plane 
parallel  to  the  2/2;-plane ;  that  x  =  a,  y  =  b  are  the  equations  of  a  line 
parallel  to  the  2;-axis ;  and  that  a;  =  a,  y  =  b,  z  =  c  represent  a  point. 
So  that  here  we  have  a  plane  represented  by  one  equation,  a  straight 
line  by  two  equations,  and  a  point  by  three. 

We  shall  now  show  that,  in  general,  one  equation  represents  a  sur- 
face of  some  kind ;  two  equations  represent  a  line  of  some  kind ;  and 
three  equations  represent  one  or  more  points. 

Let  the  equation  of  the  locus  be  F(Xj  y,  z)  =  0.  We  have  seen  that 
the  equations  of  the  line  through  the  point  (a,  b,  0)  parallel  to  the 
z-axis  are  x  =  aj  and  y  =  b.  Hence,  if  we  put  x  =  aj  and  y  =  b  in  the 
equation  of  the  locus,  we  get  the  equation  F(a,  6,  z)  =  0,  which  must 
be  satisfied  by  the  coordinates  of  all  points  common  to  this  line  and 
the  locus.  Let  the  roots  of  this  equation  be  Zi,  22,  etc.  Then  the 
locus  is  met  by  this  line  in  the  points  (a,  b,  z^),  (a,  b,  z^,  etc.  Since, 
in  general,  the  number  of  roots  of  the  equation  F{aj  bjZ)  =  0  is  finite, 
the  straight  line  will  meet  the  locus  in  a  finite  number  of  points. 
Hence  the  locus,  which  is  the  assemblage  of  all  such  points  found 
by  assigning  different  values  to  a  and  5,  is  a  surface  and  not  a  solid 
figure. 

If  the  coordinates  of  a  point  (x,  y,  z)  satisfy  two  equations 
F{x,  y^  z)  =  0  and  <^(ic,  y,  z)  =  0,  simultaneously,  the  point  must  be 
on  both  of  the  surfaces  which  these  equations  represent.  Therefore 
the  locus  is  the  curve  determined  by  the  intersection  of  the  two  sur- 
faces. When  three  equations  are  used  simultaneously,  they  are 
sufficient  to  determine  absolutely  the  values  of  the  unknown  quan- 
tities oj,  2/,  z.     Hence  three  equations  represent  one  or  more  points. 

135.  Equations  involving  only  one  or  two  variables. 

If  an  equation  contains  only  one  variable,  x  say,  let  it  be  put  in  the 
form  <l>(x)  =  0.     We  know  that  this  equation  is  equivalent  to  (x  —  a) 

200 


135]  LOCI  201 

(a?  —  6)  (x  —  c)"'  =  0,  where  a,b,c,  •••  are  roots  of  <^(«).  Hence  such 
an  equation  represents  one  or  more  planes  parallel  to  the  coordinate 
plane  x  =  0. 

Let  only  one  of  the  variables  be  absent,  so  that  the  equation  is  of 
the  form  F(x,  y)  =  0.  Let  P{x,  y,  0)  be  any  point  in  the  icy-plane 
whose  coordinates  satisfy  the  equation  F{x,  y)  =  0.  Draw  a  line 
through  P  parallel  to  the  2J-axis.  Then  all  points  on  this  line  have 
the  same  x  and  y  as  P.  That  is,  they  are  all  on  the  surface.  Hence 
the  locus  of  the  equation  F(Xj  y)  =  0  is  the  cylindrical  surface,  or 
cylinder,  traced  out  by  a  line  which  is  always  parallel  to  the  z-axis, 
and  which  moves  along  the  curve  in  the  a^-plane  defined  by  the 
equation  F(x,  y)  =  0.  In  like  manner  the  equations  /(y,  z)  =  0  and 
<li(Zj  x)=0  represent  cylinders  whose  .elements  are  parallel  to  the 
avaxis  and  y-axis,  respectively. 

If  we  treat  the  two  equations  F(x,  y,  z)  =  0  and  /(a;,  y,  2)  =  0 
simultaneously  and  eliminate  2,  we  obtain  an  equation  of  the  form 
<^(a;,  y)  =  0.  This  equation  is  satisfied  by  the  coordinates  of  all 
points  on  the  curve  represented  by  the  two  given  equations.  Since 
<^(a;,  2/)  =  0  contains  only  two  variables,  it  represents  a  cylinder 
through  this  curve  having  its  elements  perpendicular  to  the  Qcy-^\3iXiQ. 
Or,  interpreted  as  an  equation  in  plane  coordinates,  it  represents  the 
projection  of  this  curve  on  the  a;2/-plane.  Similarly,  by  eliminating  x 
and  y  we  can  find  the  projections  of  the  curve  on  the  other  two 
coordinate  planes. 

It  is  often  convenient  and  desirable  to  represent  a  curve  by  means 
of  the  equations  of  two  of  its  projecting  cylinders. 

If,  however,  we  eliminate  z  between  F{Xy  yyZ)  —  0  and  the  equation 
of  the  plane  z  =  k,  we  obtain  the  equation  F(x,  y,  k)  =  0.  This  equa- 
tion also  represents  a  projecting  cylinder  through  the  intersection  of 
the  surface  and  the  plane  z  =  k',  but  in  the  plane  z  =  0  it  represents 
a  curve  equal  in  all  respects  to  the  plane  section  of  the  surface, 
since  the  plane  of  the  section  z  =  k  is  parallel  to  the  plane  2  =  0,  on 
which  the  curve  is  projected. 

The  curves  of  intersection  of  a  surface  with  the  coordinate  planes 
are  called  the  Traces  of  the  surface.  Their  equations  may  be  found 
by  putting  x,  y,  z  in  turn  equal  to  zero  in  the  equation  of  the  surface. 
These  curves  are  very  useful  in  determining  the  nature  of  the  surface. 


202 


LOCI 


[136 


To  Trace  the  Logics  of  an  Equation 

136.  Contour  Lines.  —  A  Topographical  Map  is  one  which  gives 
not  only  the  geographical  position  of  objects  on  the  surface  of  the 
ground,  but  also  the  relative  elevations  of  the  different  parts  of  the 
surface.  On  such  a  map  the  configuration  of  the  surface  is  repre- 
sented by  means  of  Contour  Lines.  A  contour  line  is  the  projection 
on  the  plane  of  the  paper  of  the  intersection  of  a  horizontal,  or 
rather  level,  plane  with  the  surface  of  the  ground.  These  cutting 
level  planes  are  taken  5,  10,  20,  50,  or  100  feet  apart  vertically, 
beginning  with  the  datum  plane,  which  is  usually  taken  below  any 
point  in  the  surface  of  the  region  included  in  the  map. 

The  following  principles  will  assist  in  interpreting  the  meaning  of 
contour  lines :  All  points  in  one  contour  line  have  the  same  elevation 
above  the  datum  plane.  Where  ground  is  uniformly  sloping  the 
contours  must  be  equi-spaced  for  equal  changes  in  elevation,  and 
where  it  is  a  plane  they  are  also  straight  and  parallel.  In  general 
contour  lines  never  intersect  or  cross  each  other.  Two  exceptions  to 
this  rule  should  be  carefully  noted,  viz.  a  contour  line  will  cross  itself 
at  a  pass,  they  cross  each  other  at  overhanging  precipices.  Every  con- 
tour line  must  either  close  upon  itself  or  extend  continuously  across 
the  map.  Where  a  contour  line  closes  upon  itself  the  included  area 
is  either  a  hill-top  or  a  depression  without  an  outlet. 


What  is  the  nature  of  the  surface  shown  by  the  contour  lines  in  this  figure  ? 


13a]  LOCI  203 

It  is  obvious  that  this  method  of  contours  can  be  used  to  determine 
the  general  nature  of  the  surface  represented  by  any  given  equation 
F{Xy  y,  z)  =  0.  If  we  put  z  =  k  in  this  equation  we  get  the  equation 
F(x,  y,  A;)  =  0,  which  represents  the  projection  on  the  plane  2  =  0  of. 
any  plane  section  of  the  given  surface  parallel  to  this  coordinate  plane. 
By  assigning  different  values  to  k  we  can  get  as  many  such  sections  as 
we  choose.  In  like  manner,  by  putting  a;  =  A;,  and  y  =  k,we  can  find 
sections  parallel  to  the  other  coordinate  planes.  We  may  for  con- 
venience call  these  sections  contours  of  the  given  surface.  These  three 
systems  of  contours  will  indicate  the  general  nature  of  the  surface. 

Find  the  contours  of  the  surfaces  whose  equations  are 

I.   x-\-y  +  z  =  l.  2.   a;2  4. 2,2  4.  ^2  =  a2.  3.   x^  +  y2  =  c^. 

137.  An  equation  of  the  first  degree  represents  a  plane. 
The  most  general  equation  of  the  first  degree  is 

Ax-{-By-^Cz-{-D  =  0.  (1) 

If  we  put  z  =  kin  this  equation,  we  get 

Ax  +  By+Ck-{-D=:Oy  (2) 

which  for  different  values  of  k  represents  a  system  of  parallel 
straight  lines.  The  contours  on  the  planes  yz  and  zx  are  also  par- 
allel straight  lines. 

The  distance  between  the  two  contours  made  by  the  planes  z  =  ki 

and  2J  =  /cg  is       ]       ^^ .     But  this  distance  varies  directly  as  (fcj  —  kX 

V-4^  +  B^ 
the  distance  between  the  two  planes  z  =  kx  and  2  =  ^2.     Therefore 
equation  (1)  represents  a  plane. 

138.  Trace  the  surface  represented  by  the  equation 

x'-{.fj^z^-{.2Ax  +  2By-\-2Cz-{-D  =  0.  (1) 

The  jcy-contours  of  this  surface  are  the  concentric  circles 

a?  +  f  +  2Ax-\-2By  +  k^  +  2Ck-^D=:0,  (2) 

with  centres  at  (—  ^,  —B)  and  radii  equal  to 

V^^  +  ^-(Ar*-f.2Cfc4-i>), 


which  become  zero  if  k  =  —C±  V-4'  -\-  B^  -\-  C^  —  D^  and  imaginary 


if  k>-C+^/A^  +  EP-^C'-D, 

or  if  kK-C-^/A'-^B'-^-C-D. 


204 


LOCI 


[139 


The  a»-coiitours  are  circles  whose  equations  may  be  written 

(x-\-Ay-\-(z-\-Cy  =  A'+C'-(J<^  +  2Bk  +  D).  (3) 

Likewise  the  2/2!-contours  are  circles  whose  equations  are 

(y-{-By-{-(z+Cy  =  B'+C'-{lc'-{-2AJc-\-B).  (4) 

Moreover,  the  centres  of  these  three  systems  of  concentric  circles 
are  the  projections  of  the  point  (—A,  —  B,  —  C),  and  the  radius  of 
each  system  is  -y/A^  -\-B^-{-C^  —  D  when  k  is  equal  respectively  to 
—  O,  —B,  and  —  A.  Hence  these  contours  indicate  that  the  sur- 
face is  a  sphere  with  centre  at  the  point  (—A,  —By  —  C)  and  radius 
equal  to  V^^  4-  ^  +  C^  —  D.  This  can  be  shown  to  be  true  by 
writing  the  given  equation  (1)  in  the  form 

(x-^Ay-\-{y->rBy  +  {z  +  Cy  =  A'  +  B'-{-C'-D,  (5) 

and  comparing  with  equation  (1)  §  129. 

Hence  the  equation  of  the  sphere  whose  centre  is  the  point  (a,  6,  c) 

and  radius  r  is 

{nc  -  a)2  +  (y  -  &)2  +  (s  -  C)2  =  r2.  (6) 

If  the  centre  is  at  the  origin,  the  equation  is 

i»2  +  2/2  +  s2  =  r2.  (7) 

139.   Trace  the  surface  whose  equation  is    [Frost's  S.  G.  p.  5.] 

{x  +  yy  =  az. 

When  x  =  0,y^  =  az;  therefore  the  trace  on  the  2/2;-plane  is  a  parabola  OQ, 

whose  axis  is  OZ  and  vertex  O. 

Similarly  the  trace  OP  on  the  x^-plane  is 
the  equal  parabola  x^  =  az,  having  the  same 
vertex  and  axis. 

If  z  =  Jc,  (x  +  yy  =  ak.  That  is,  any  xy- 
contour  is  two  parallel  straight  lines,  equally 
inclined  to  the  x  and  y-axes. 

Hence  the  surface  is  a  cylinder  generated 
by  a  straight  line  PQ  moving  along  the  two 
equal  parabolas  y"^  =  az  and  x^  =  az,  and 
always  parallel  to  the  straight  line  a;  +  y  =  0 
in  the  cc?/-plane. 

The  other  two  systems  of  contours  are 
parabolas  which  are  all  equal  to  the  traces 
OP  and  OQ. 


140]  LOCI  205 

EXAMPLES 
Trace  the  surfaces  represented  by  the  following  equations : 

1.  2x-^Sy-4z  =  12.  ^^    ^^t-^^i 

2.  X2  +  2/2  +  ;j2  =  16.  ■     a2       52        c2  ' 

3.  x^  +  y^  +  z^-ix  +  6y-2z  =  ll.  15.  ^_l^_?!  =  i 

4.  .2  +  ,2  =  «2.  «^     ft^     c2       • 
5.y^^z^  =  2az.  ''  (x^yy^z^^a^ 
B.y^  =  4az.  1^-  ^^  =  2... 

7.  ;22_y2  =  «2.  18.    (aj  +  y)2  =  2(a2-02). 

8.  x'^+y^  =  z\  19-    (a;  +  y  -  a)2  +  2;2  =  a2. 

9.  X2  +  2/2  =  <j;j.  20.     (X  -  2)2  +  (1/  -  0)2  =  a2. 

10.   2^  +  ?'  =  4a;2.  21-    («  +  2/)^  =  c(;3  -  a;). 

*'*       ^^  22.     C2y2  =  a.2(«2  _  ;22). 

a;2       w2  jr  \ 

^1-   ^2+52  =  ^^-  23-  a;02  =  c22,. 

12.   z^  =  ax+by.  24.  a;y  =  a^. 

j3    ^     y2^_j  25.  y  =  xUnz. 

«^     ft'^     c2  26.  xyz  =  a. 

Show  that  the  following  pairs  of  equations  represent  the  same  locus,  and 
trace  their  loci : 

27.  p  =  a  cos  0  and  x^  +  y^-\-  z^  =  az. 

28.  p  =  a  sin  0  and  (pfi  +  y^  +  ^2)2  _  ^2(jc2  ^y'^), 
2B.  p  =  a  cos  0  and  (x^  +  y"^  +  z"^)  {x^  +  2/2)  =  ^{23.2. 
80.  p  =  a  sin  0  and  (ajS  +  2/^  +  2;2)  (a;2  +  2/^)  =  a^y\ 

To  Find  the  Equation  of  a  Locus 

140.  If  a  point  moves  in  space  subject  to  a  given  condition,  it  will 
generate  a  locus.  This  locus  is  the  totality  of  positions  the  point  may 
have  under  the  given  condition.  For  example,  a  point  keeping  at  a 
constant  distance  from  a  fixed  plane  will  generate  a  parallel  plane ; 
a  point  keeping  at  a  constant  distance  from  a  fixed  straight  line  will 
generate  a  cylinder.  If  we  can  find,  in  any  system  of  coordinates, 
an  algebraic  equation  that  is  satisfied  by  the  coordinates  of  every 
point  on  the  locus,  and  not  satisfied  by  the  coordinates  of  any  other 
point,  we  shall  have,  as  in  plane  geometry,  the  equation  of  the  locus. 


206 


LOCI 


[141 


In  the  second  example  just  cited,  for  instance,  if  the  2;-axis  is  taken 
as  the  fixed  line  and  a  as  the  constant  distance,  the  equation  of  the 
locus  will  be  a^  +  2/^  =  a^ ;  for  this  equation  is  satisfied  by  the  coordi- 
nates of  any  point  (x,  y,  z)  whose  distance  from  the  z-axis  is  a,  and 
by  no  other  point. 

In  finding  the  equation  of  a  locus  in  space,  the  general  method  of 
procedure  is  the  same  as  in  plane  geometry. 


Surfaces  of  Kevolution 

141.  A  Surface  of  Revolution  is  a  surface  generated  by  revolving 
a  plane  curve  around  a  fixed  line  in  the  plane  of  the  curve. 

To  find  the  general  equation  of  a  surface  of  revolution  we  will  take 

the  aj-axis  for  the  fixed  line,  and  let  the  equation  of  the  generating 

curve  be  ^,  ^ 

y^fix).  (1) 

Any  point  P  on  the  generating 
curve  AB  will  describe  a  circle  whose 
plane  is  perpendicular  to  the  jc-axis, 
and  whose  radius  is  CPy  the  ordinate 
of  the  generating  curve.  Hence  for 
every  point  Pix,  y,  z)  on  this  circle 

^^^^^^^       f  +  ,^=OP\  (2) 

For  all  positions  of  P  on  the  gen- 
erating curve 

CP=f(x).  (3) 

Therefore  the  required  equation  is 

2/2  +  «2  =  [/(a?)]2.  (4) 

Similarly,  the  equations  of  surfaces  of  revolution  about  the  other 
two  coordinate  axes  are 


a;2  +  2j2  =  [/(i,)]2  and  a;2  +  2/2  =  [/(2;)]2, 


(^ 


For  example,  if  the  circle  ix^-\-y^  =  7^  is  revolved  about  the  a:-axis, 
we  have  CP  =  Vr^  —  x^  =  f(x) ;  and  the  equation  of  the  generated 
sphere  is  x^ -^  y^ -\' z^  =  r^. 


141]  LOCI  207 

Likewise  the  equation  of  the  cone  generated  by  revolving  the  line 
y  =  mx  about  the  ic-axis  is 

1/2 +  «2:=  ^2^.2.  (6) 

If  we  eliminate  z  between  this  equation  (6)  and  the  equation  of  the 
plane  z  =  m'x  +  c  we  get  for  the  projection  of  the  conic  section  on  the 
iB^-plane  ^^  ^  ^^n  _  ^2)  ^.2  ^  2  cm'x  +  c^  =  0.  (7) 

Show  that  this  section  is  an  ellipse,  a  parabola,  or  a  hyperbola 
according  as  m'>,  =,  or  <m;  and  that  if  c  =  0,  it  is  either  a  point, 
two  coincident  lines,  or  two  intersecting  lines.  What  property  of  the 
conic  section  does  this  prove  ? 

EXAMPLES 

1.  Show  that  the  locus  of  all  points  in  space  equally  distant  from  the  two 
points  (3,  —  2,  1)  and  (  —  2,  1,  —  3)  is  the  plane  5a;  —  3y  +  45?  =  0. 

2.  Show  that  all  points  which  are  equidistant  from  the  three  points 
(4,  —  1,  —  2),  (  —  2,  4,  —  1),  and  (—1,  —  2,  4)  are  on  the  line  whose  equations 
are  x:=y  =  z. 

3.  A  point  moves  so  that  its  distance  from  the  origin  is  twice  its  distance 
from  the  plane  z  =  0.    Find  the  locus  of  the  point.  Ans.  x^-\-y^  =  S  z^. 

4.  Find  the  locus  of  a  point  which  moves  so  that  (1)  the  sum,  (2)  the  dif- 
ference of  the  squares  of  its  distances  from  the  points  (a,  0,  0)  and  (—a,  0,  0) 
is  the  constant  2  c^. 

6.  Find  the  locus  of  a  point  such  that  the  sum  of  the  squares  of  its  distances 
from  the  three  points  (3,  -  3,  5),  (-  1,  1,  -  2),  and  (4,  2,  -  3)  is  38. 

Ans.  ic2  _|.  y2  +  2-2  _  4  jc  _|_  26  =  0. 

6.  Show  that  the  locus  of  a  point  the  sum  of  the  squares  of  whose  distances 
from  n  fixed  points  is  constant  is  a  sphere. 

7.  Find  the  locus  of  a  point  such  that  the  sum  of  the  squares  of  its  distances 
from  the  faces  of  a  cube  is  constant. 

8.  Find  the  equations  of  the  surfaces  of  revolution  generated  by  revolving 
the  conic  sections  around  their  axes. 

9.  Find  the  equation  of  the  surface  generated  by  revolving  the  parabola 
around  the  tangent  at  the  vertex. 

10.  Find  the  locus  of  a  point  which  moves  so  that  its  distance  from  the  point 
(2  a,  0,  0)  is  always  equal  to  its  distance  from  the  plane  cc  =  0. 


208  LOCI  [141 

11.  Find  the  locus  of  a  point  such  that  (1)  the  sum,  and  (2)  the  difference,  of 
its  distances  from  the  two  points  (c,  0,  0)  and  (—  c,  0,  0)  is  constant  and  equal 
to  2  a. 

12.  Show  that  the  equation  of  the  surface  generated  by  revolving  the  circle 
a;2  ^.  2;2  =  2  ax  around  the  2;-axis  is 

(a:2  +  2/2  +  22.)2  =  4  ^2  (a;2  +  2/2) . 

Show  also  that  the  equation  in  spherical  coordinates  is 

p  =  2  a  sine.  (See  Ex.  28,  p.  205.) 

13.  The  six  points  ^(a,0,0),5(- a,  0,0),  C(0,  a,0),  Z>(0,  -a,  0),  ^(0,0,  a), 
and  F(0,  0,  —  a)  form  a  regular  octahedron. 

Find  the  locus  of  a  point  P  in  space  such  that 

(1)  ^P2  +  5P2  +  ^P2=CP2  +  2)P2  +  i?'P2;    Ans.z  =  0. 

(2)  u4P2+C'P2  +  JSrP2  =  PP2  +  2)p2  +  PP2;    Ans.x  +  y-\-z  =  0. 

(3)  ^P2+C'P2=PP2+2)p2+^p2+i^p2.     Ans,  x'^+y^-\-z'^+2a(x+y)+a'^=0, 

(4)  ^p2  +  PP2=OP2  +  i)P2+^p2  +  pp2;    Ans.  z^  +  y^  +  z^  +  tt"^  =  0. 

(5)  ^P2  +  PP2  =01^  +  DI^=  EP^  +  PP2 ;    Ans.  All  space. 

14.  If  ABCD  is  a  regular  tetrahedron,  show  that  the  locus  of  a  point  P,  such 
that  2  P^2  _  pj52  _}.  pc'2  +  P2)2^  ig  a  sphere  passing  through  the  points,  P,  0,  D, 
and  having  a  radius  equal  to  twice  the  face  altitude  of  the  tetrahedron. 

15.  Show  that  the  equation  8  +  ^8'  —  0  represents  a  surface  passing  through 
all  the  common  points  of  the  two  surfaces  ^S'  =  0  and  8^  —  0.  Show  also  that 
88'  —  0  represents  both  of  the  surfaces  /S'  =  0  and  8'  —  0. 

16.  Find  the  equation  of  the  surface  of  the  blade  of  a  screw-auger. 


CHAPTER  XIV 
THE  PLANE  AND  THE  STRAIGHT  LINE 
142.    To  Jind  the  equation  of  a  plane. 


Let  OH  be  perpendicular  to  the  given  plane  ABC,  intersecting  it 
in  jfiT;  and  let  I,  m,  n  be  the  direction  cosines  of  OH.  Let  OK=p 
be  the  distance  measured  from  the  ongin  to  the  plane,  and  let  P(xj  y,  z) 
be  any  point  in  the  plane.  Draw  PR  perpendicular  to  the  plane 
XOFand  BQ  perpendicular  to  OX. 

Then  OK,  the  projection  of  OP  on  OH,  is  equal  to  the  sum  of  the 
projections  of  OQ,  QB,  and  MP  on  OH  [(1),  §  130] 

Therefore  Ix  +  my  ■\-nz  =  Pf  (1) 

which  is  the  equation  of  the  plane  in  the  Normal  or  Distance  Foi-m. 

Since  changing  the  signs  of  all  its  direction  cosines  reverses  the 
direction  of  a  line,  the  equation  of  a  plane  may  always  be  written 
so  that  p  shall  be  measured  along  the  positive  direction  of  OH-, 
i.e.  so  that  p  shall  be  positive.    The  positive  side  of  the  plane  is 

209 


210  THE  PLANE  AND  THE  STRAIGHT  LINE  [143 

found  by  going  from  the  plane  in  the  positive  direction  of  p.     Hence 
when  p  is  positive  the  origin  is  on  the  negative  side  of  the  plane. 
Equation  (1)  may  also  be  written  in  the  form 

^  +  i^  +  ^  =  l.  (2) 

P         P  P 

I      m     n 

T)  T)  If) 

If  now  we  let  a  =  ^,  6  =  — ,  and  c  =  -,  we  have 
V        m  n' 

which  is  the  equation  of  the  plane  in  terms  of  its  intercepts  on  the 
Les. 
The  general  equation  of  the  first  degree 


may  be  written 

By          ^            Cz           _ 

-D 

(4) 

='  (5) 

V^2  +  _B2+C2       V^2  +  jB2_,.Cf2       V^2  +  ^2+(72       -^A^  +  B^+C^ 

in  which  the  coefficients  of  a;,  y,  and  z  are  the  direction  cosines  of 
some  line  [(5),  §  131].  Comparing  this  with  equation  (1)  we  see 
that  (5)  is  the  equation  of  a  plane  in  the  distance  form. 

143.    The  distance  from  a  given  plane  to  a  given  point. 

The  demonstration  is  precisely  the  same  as  that  for  the  corre- 
sponding proposition  in  Plane  Geometry. 

If  d  represents  the  distance  and  (x^,  y^  z^  is  the  given  point,  the 
required  formula  is 

d  =  lxi  +  mm  +  nzi  -p,  or  g^^xi^  Byi  + Czi  + D^        .^ 

according  as  the  equation  of  the  plane  is 

lx-\-my-\-  nz  =py  or  Ax  +  By  +  Cz-\-D  =  0. 

As  in  Plane  Geometry  a  point  (x'j  y\  z')  is  on  the  positive  or 
negative  side  of  the  plane  Ax  -{-  By  -{-  Cz  -\-  D  =  0,  according  as 
Ax'  +  By'  -\-Cz'  -{-D  is  positive  or  negative. 


X, 

y, 

2^, 

1 

Xi, 

Vh 

«1, 

1 

X2, 

^2, 

5?2, 

1 

X3, 

2/3, 

^3, 

1 

144]  THE  PLANE   AND  THE  STRAIGHT  LINE  211 

EXAMPLES 

1.   Show  that  the  equation  of  a  plane  through  the  three  points  (xi,  yi,  01), 
(3^2,  2/2,  22),  and  (X3,  2/3,  5?3)  is 


=  0. 


2.  Find  the  equation  of  the  plane  through  the  three  points  (1,  2,  2), 
(2,  —4,  —3),  and  (—  6,  2,  5).     Find  j9,  the  intercepts,  and  traces  of  the  plane. 

Ans.  2x  —  3y  +  42;  =  4. 

3.  Find  the  equation  of  the  plane  through  the  point  (3,  2,  —  4)  parallel  to 
the  plane  2x  —  3y  —  5^  =  0.  Ans.  2x  —  Sj/  —  6^  =  20. 

4.  If  >S^  =  0  and  S'  =  0  are  the  equations  of  two  planes,  show  that  S+\S'  =  0 
will  be  the  general  equation  of  a  plane  through  their  intersection. 

5.  Find  the  equation  of  a  plane  through  the  origin  and  through  the  inter- 
section of  the  two  planes  3x  +  4y  —  2^  +  4  =  0  and  4x  —  5x  —  ^  =  6. 

Ans.  nx-\-2y-Sz  =  0. 

6.  Show  that  the  four  planes  x  —  y  —  2z  =  lj  2x  — 2/4-2  +  1  =  0,  x-\-2y  —  z  =  6, 
and  4x4-y  +  60  =  O  meet  in  a  point. 

Find  the  general  condition  that  four  planes  shall  meet  in  a  point. 

7.  Show  that  the  four  points  (0,  1,  3),  (1,  1,  1),  (-2,  -3,  -5),  and 
(4,  2,  —2)  are  in  the  same  plane.     (Use  the  determinant  in  Ex.  1.) 

8.  Show  that  the  two  points  (1,  —4,  —2)  and  (—1,  2,  3)  are  on  opposite 
sides  of  the  plane  7x  —  3«/  +  42;  =  5,  and  equidistant  from  it. 

9.  Show  that  the  equations  of  the  planes  which  bisect  the  angles  between  the 
two  planes  Az  +  By  +  Cz -]-  D  =  0  and  A'x  +  B'y  +  C'z  +  Z>'  =  0, 

are  Ax  -h  By  -[■  Cz  ■}-  D  ^  ^A'x  -^-  B'y  +  C'z  +  D' 

V^2  +  ^  +  02  V^'2  4.  B'^  +  C'^ 

144.   Equations  of  a  straight  line^ 

We  have  seen  in  §  134  that  it  requires  two  equations  used  simul- 
taneously to  represent  a  line  in  space.  Since  two  planes  intersect 
in  a  straight  line  we  may  take  the  two  general  equations  of  the 
first  degree 

Ax-{-By  +  Cz-^D  =  Oy  and  A'x  + B'y-{-C'z-\-D' =  0,        (1) 

as  the  most  general  equations  of  a  straight  line. 
If  we  treat  these  equations  simultaneously  and  eliminate  2,  y,  x, 


212 


THE  PLANE  AND  THE  STRAIGHT  LINE 


[145 


respectively,  we  obtain  three  other  consistent  equations  which  may 
be  reduced  to  the  form 


b~^'  h'  '  c~"'  c'  '  a'~"  (^) 

Since  each  of  these  equations  (2)  is  satisfied  by  the  coordinates  of 
every  point  on  the  line,  they  will  each  determine  a  plane  through 
the  line.  These  planes  are  seen  to  be  the  projecting  planes  of  the 
line,  while  their  equations  also  represent  the  projections  of  the  line 
on  the  coordinate  planes.  The  equations  of  any  two  of  the  project- 
ing planes  may  be  chosen  as  the  equations  of  the  line. 

If  the  line  is  parallel  to  one  of  the  coordinate  planes,  two  of  the 
projecting  planes  coincide  and  the  equations  of  the  line  will  be  of 
the  form  bx  -\-  ay  =  ab,  z  =  c;  if  the  line  is  parallel  to  one  of  the 
axes,  one  of  the  projecting  planes  is  indeterminate,  and  the  other 
two  are  of  the  form  x  =  a,  y=b. 

From  the  equations  (2)  of  the  projecting  planes  we  see  that  the 
coordinates  of  the  points  where  the  line  meets  the  coordinate  planes 
x  =  0,  2/  =  0,  z  =  Oj  are  respectively  (0,  6,  0%  (a,  0,  c),  (a',  b',  0). 

The  equations  of  a  straight  line  contain  four  independent  constants. 

145.    The  symmetrical  equations  of  a  straight  line. 
Let  P'(x',  y',  z')he  a  fixed  point  on  the  line,  and  P(Xj  y,  z)  any 
other  point  on  the  line  at  a  distance  r  from  P' ;  let  I,  m,  n  be  the 

direction  cosines  of  the  line  P'P. 
Through  P '  and  P  draw  planes 
parallel  to  the  coordinate  planes, 
making  a  parallelopiped  whose 
edges  P'Q,  QR,  and  MP  are 
respectively  equal  to  the  projec- 
tions of  P'P,  or  r,  on  the  axes. 
Since  these  edges  are  respectively 
equal  to  x  —  x',  y  —  y',  and  z  —  z\ 
we  have 
aj  —  «'  =  Zr,  y  —  y^  —  mr,  z  —  z^  —  nr,  (§  130)     (1) 


or 


a?-ac' 


I  m,  n 

which  are  the  required  equations  of  the  line. 


147]  THE  PLANE  AND  THE   STRAIGHT  LINE  213 

146.  To  find  the  equations  of  a  straight  line  through  two  given 
points  (»!,  yi,  z^  and  (a^g,  Vz)  ^2)- 

Since  the  line  passes  through  the  point  (ajj,  y^  Zi)  its  equations 
will  be  of  the  form  [(2),  §  145] 

I  m  n 

Then,  since  the  point  (ajg,  2/2?  ^2)  is  also  oli  the  line,  we  have 

X2-  xi  _y2-  yi  ^Z2-  Zi  ^2) 

I  m  n 


Dividing  (1)  by  (2)  gives  the  required  equations, 
x-xi      y-vi      z-zx 


(3) 


a?2-a?i     y^-vt    z^-zi 

Hence,  the  direction  cosines  of  the  line  are  proportional  to  the 
differences  of  the  coordinates  of  the  two  given  points. 

147.  The  equations  of  any  two  straight  lines  in  rectangular  co- 
ordinates can  be  written  in  a  very  simple  form  by  a  proper  choice  of 
axes. 

Take  the  middle  point  of  the  shortest  distance  between  the  two 
lines  for  the  origin,  and  the  z-axis  along  this  line.  Take  the  yz  and 
xz  planes  so  that  they  bisect  the  angles  between  the  two  planes 
determined  by  the  z-axis  and  the  two  given  lines.  Then  the  equations 
of  the  two  lines  can  be  written 

y  =  mx,  z  =  Ct  and  y  =  -mx,  s  =  -c,  (1) 

or  in  the  symmetric  form 

EXAMPLES 
1.   Find  the  symmetric  equations,  and  the  direction  cosines,  of  the  line  of 
intersection  of  the  planes  6x  —  y-\-z  +  6  =  0  and  x  —  y  —  z-\-l  =  0. 
Eliminating  z  and  y  in  turn  between  these  equations,  we  get 
Zx  =  y-S  and  2x-{-z-\-2  =  0. 

Whence  ^  =  1L^  =  ^.±1. 

13-2 

Hence  the  direction  cosines  of  the  line  are  proportional  to  1,  3,  and  —  2 ;  and 

1  o        2 

their  actual  values  are ,   , 

Vli    Vli    V^lT 


214 


THE  PLANE  AND  THE  STRAIGHT  LINE 


[148 


Find  the  projections,  the  symmetric  equations,  the  points  where  they  pierce 
the  coordinate  planes,  and  the  direction  cosines  of  the  lines  whose  equations  are 

2.  x-{-y-z  +  l=0  and  ix  +  y  +  z  =  5. 

3.  x  +  y-z  +  l  =  0  and  ^x-\-y-2z-\-2  =  0. 

4.  2x-y-\-z-S  =  0        and  x-{-2y  +  z  =  5. 

5.  8x-2y  +  ^z  =  12       and  6 x - 4 y - S z -{■  2i  =  0. 

6.  5x-Sy  +  2z-{-6  =  0  and  Sx-5y-2z  =  7. 

7.  Write  the  symmetric  equations  of  a  line  perpendicular  to  a  coordinate 
axis;  a  coordinate  plane. 

8.  Write  the  symmetric  equations  of  the  line  through  the  point  (2,  —  3,  1) 
equally  inclined  to  the  axes. 

9.  Find  the  equations  and  direction  cosines  of  the  line  through  the  two 
points  (- 1,  3,  2)  and  (2,  -  3,  0). 

10.  Find  the  equations  of  the  line  through  the  origin  perpendicular  to  the 
plane  lx  +  my  +  nz=  p. 


11.   Find  the  coordinates  of  the  point  where  the  line 
meets  the  plane  2a;  —  y  —  355  +  15  =  0. 


x-2 

1 


y  -  2  ^  g  +  3 

-2        -3 


148.    To  find  the  angle  between  two  straight  lines  whose  direction 

cosines  are  given. 

Draw  OP  and  OP  through  the 
origin  parallel  respectively  to  the 
two  given  lines.  Let  l,  m,  n  and 
V,  m'j  n'  be  the  direction  cosines 
of  OP  and  OP'  respectively,  and 
let  6  represent  the  angle  POP'. 

Let  p  be  the  distance  from  the 
origin  to  the  point  P{xy  y,  z). 
Then  projecting  p,  x,  y,  and  z  on 
OP'  we  get  [(1),  §  130] 

p  cos  d  =  I'x  4-  m'y  +  n'z.  (1) 

But  x^lp,  y  =  mp,  and  z  =  np.     .    [(2),  §  131.]    (2) 

.-.  cos9  =  ir -\-tnni'-\-nn'.  (3) 

It  0  =  90°,  cos  d  =  0.    Hence  the  condition  for  perpendicularity  is 

W  +  mm'  +  nn'  =  0.  (4) 


149]  THE  PLANE   AND  THE  STRAIGHT  LINE  215 

It  should  be  noticed  that  equation  (3)  gives  the  angle  between  two 
lines  directed  from  the  origin.  If  the  signs  of  Z,  m,  n  are  all  changed, 
the  direction  of  OP  will  be  reversed,  the  sign  of  IV  +  mm' -\- nn'  will 
be  changed,  and  0  will  be  the  supplement  of  its  former  value.  But  if 
the  signs  of  V  m'  n'  are  also  changed,  the  direction  of  both  lines  will 
be  reversed,  the  sign  of  cos  0  will  not  be  changed,  and  6  will  be  un- 
altered. 

149.    To  find  the  angle  between  two  planes. 

The  angles  between  two  planes  are  evidently  equal  to  the  angles 
between  the  lines  through  the  origin  perpendicular  to  the  planes. 
Let  the  equations  of  the  planes  in  the  distance  form  be 

Ix  +  my -\- nz  =  py  (1) 

and  l'x-{-m'y-{-n'z=p',  (2) 

Then  cos  6  =  W  +  mn'  +  nn'.      [(3),  §  148.]  (3) 

If  the  planes  are  at  right  angles,  cos  ^  =  0 ;  i.e. 

IV  +  mm'  +  nn'  =  0.  (4) 

If  l  =  V,  m  =  m',  and  n=:n',  then  cos  0  =  1,  and  the  planes  are 
parallel.  ^ 

If  the  equations  of  the  planes  are 

Ax  +  By-^Cz  +  D  =  Oy  (5) 

and  •    A'x  +  B'y-\-az-\-D'  =  0,  (6) 

eos9=  AA'  +  BB'  +  Ca         _,     [(5),  §  142]     (7) 

V^2  +  ^2  +  (72  .  VJL'2  +  JB'2  +  Cf'2 

and  the  condition  for  perpendicularity  is 

AA'  +  BB'  +  CC'  =  0.  '  (8) 

If  ^  =  kA',  B  =  kB'  and  C=  kC,  the  planes  are  parallel. 

Let  the  equations  (1)  and  (2)  be  written  so  thatp  and  jp'  have  the 
same  sign.  Then,  when  cos  6  is  positive  the  angle  between  p  and  p' 
is  acute,  and  the  angle  between  the  planes  in  which  the  origin  lies  is 
obtuse.  If  cos  6  is  negative^  the  origin  lies  in  the  a^mte  angle  between 
the  planes. 


216  THE  PLANE  AND  THE   STRAIGHT  LINE  [149 


EXAMPLES 

1.  Show  that  the  lines  4x  =  — ?/  =  3;s  and  Sx  =  —  4ty  =  —  z  are  perpen- 
dicular to  each  other. 

2.  Find  the  angle  between  the  lines  

Ans.  cos-^ . 

26 

3.  Find  the  angle  between  any  two  of  the  four  lines  through  the  origin 
equally  inclined  to  the  axes. 

4.  Find  the  angle  between  any  two  of  the  lines  which  bisect  the  angles 
between  the  axes. 

5.  Find  the  angle  between  one  of  the  lines  in  Ex.  3  and  one  of  the  lines 
in  Ex.  4. 

6.  Find  the  angle  between  the  planes  x  +  y-\-  z  =  1  and  x  —  y  —  2z  =  2. 

Is  the  origin  in  the  acute  or  the  obtuse  angle  ?  the  point  (1,  3,  —  1)? 

V2 
Ans.  cos-i  — — ,  Acute,  Obtuse. 

o 

7.  Find    the    equation    of    the    plane    through    the    line     x-\-  y  —  z  =  2j 
2x  —  Sy-\-4:Z-\-6  =  0  and  perpendicular  to  the  plane  x  —  2y  +  z  =  0. 

Ans.  Sx-\-Sy  -2z  =  7. 

8.  Find  the  equation  of  the  line  througji  the  point  (1,  4,  3)  perpendicular  to 
the  plane  3x  —  2^  +  42;  =  0. 

9.  Find  the  equation  of  the  plane  through  the  point  (2,  —1,  3)  and  perpen- 
dicular to  the  line  2x  +  Sy  —  z=2,  x-2y  +  z  =  S. 

Ans.  x-Sy-7z  +  lQ  =  0. 

10.  Find  the  dihedral  angles  of  a  regular  octahedron. 

11.  Show  that  the  line  -  =  —  =  -  will  be  parallel  to  the  plane 

I      m     n 

I'x  4-  tn'y  +  n'z  =p,  if  W  +  mm'  +  nn'  =  0. 

12.  Show  that  the  equations  of  the  straight  lines  which  bisect  the  angles 
between  the  lines 

?  =  l  =  i   and   |  =  J^  =  1. 
I     m     n  V     m'     n' 

are  ^-^V^^   and      *    -      ^      -     '^ 


l  +  V     w  -r  m'     w  +  n'  l  —  V     m  —  m'     n  —  n' 


151] 


THE  PLANE  AND  THE  STRAIGHT  LINE 


217 


Transformation  op  Coordinates 

150.  To  change  the  origin  of  coordinates  loithout  changing  the  direc- 
tion of  the  axes. 

This  transformation  is  evidently  similar  to  the  corresponding  one 
in  Plane  Geometry.  Hence  if  we  wish  to  find  the  equation  of  a 
locus  referred  to  new  axes  parallel  respectively  to  the  old,  and 
passing  through  the  point  (ajo,  2/o?  ^o))  we  have  only  to  write  in  the 
place  of  X,  2/,  2;,  respectively, 

151.  To  change  the  direction  of  the  axes  without  changing  the  origin. 

Let  Zi,  mi,  «i ;  Zg?  ^2j  ^2;  a,nd  Zg,  mg,  %,  be  the  direction  cosines  of 
the  new  axes  OX',  0  Y^,  OZ'  respectively,  referred  to  the  old  axes. 

Let  P  be  any  point  (a;,  y,  z)  referred  to  the  old  axes,  and  let  its 
coordinates  referred  to  the  new  axes  be  OQ  =  a;',  QR  =  y\  RP  =  z'. 

Then  projecting  the  lines  OP,  x', 
y',  z'  on  the  old  axes  OX,  OT,  OZ, 
respectively,  we  get  [(1),  §  130] 

y  =  mtoc'  +  m^y'  +  mgs',  I  (1) 
and  z  =  n\x'  +  n^y'  +  ns^'.     J 

These  are  the  required  formulae. 

The  student  should  compare 
them  with  the  corresponding  for- 
mulae in  Plane  Geometry. 

It  is  evident  that  the  degree  of 
an  equation  will  not  be  altered  by 
either  of  these  transformations. 

The  direction  cosines  of  the  old  axes  referred  to  the  new  are 
respectively  l^,  I2,  /g ;  m^  m^,  m^ ;  and  Wj,  712,  Wg. 

Hence  we  have  the  six  relations 

h^  +  mi^  +  V  =  1,  ] 


(2) 


218 


THE  PLANE  AND  THE  STRAIGHT  LINE 


[151 


n^  +  rii  +  ni  =  1.  J 


(3) 


Since  both  sets  of  axes  are  rectangular,  we  also  have  the  six 
equations 

1^2  -H  m-i^mi  +  921^2  =  0, 

Uz  +  wigms  +  W2W3  =  0,  •  (4) 

^1  +  Wgrni  +  Tiarii  =  0 ;  - 

liTTix  -h  Zgma  +  ?3m3  =  0, 

miWi  +  mgWa  4-  mgWg  =  0,  (5) 

Wi^i  +  712^2  +  WgZg  =  0. 


EXAMPLES  ON  CHAPTER  XIV 

1.  Transform  the  equation  (x  +  yy^  =  az  by  turning  the  axes  of  x  and  y 
around  the  0-axis  through  an  angle  of  45^.  Ans.  2x^  =  az, 

2.  If  P  is  a  fixed  point  on  a  straight  line  through  the  origin  equally  inclined 
to  the  axes,  any  plane  through  P  will  intercept  lengths  on  the  axes  the  sum  of 
whose  reciprocals  is  constant. 

3.  The  equation  of  the  plane  through  the  line  -  =  ^  =  -,  and  which  is  per- 

l      m     n 

pendicular  to  the  plane  containing  the  lines  —  =  ^  =  ?  and  -  =  ^  =  — ,  is 

m     n     I         n      I     m 

(m  —  n')x  +  (w  —  X)y  +  (Z  —  m)z  —  0. 

4.  Show  that  the  three  straight  lines 

x_'f_z       x_y_z       ^-.y_-.^ 
a^y^abc^lmn 
Will  lie  in  one  plane  if 

a{bn  —  cm)  +  /3(d  —  an)  +  y{am  —^l)  =  0. 

6.  If  a,  6,  c  and  a',  b',  d  are  the  intercepts  of  a  plane  on  two  sets  of  rectan- 
gular axes  having  the  same  origin,  then 

i  +  l  +  l  =  i  +  -l  +  l. 
a2      62      c2      a'2      &'2  ^  c'2 

6.  The  locus  of  a  point  whose  distances  from  two  given  planes  are  in  a 
constant  ratio  is  a  plane. 


151]  THE  PLANE   AND  THE  STRAIGHT  LINE  219 

7.  Show  that  the  locus  of  a  point  which  moves  so  that  the  sum  of  its  distances 
from  two  fixed  planes  is  constant  is  a  plane  parallel  to  one  of  the  planes  which 
bisect  the  angles  between  the  two  fixed  planes.  "What  is  the  locus  if  the  difference 
of  these  distances  is  constant  ? 

8.  Find  the  locus  of  a  point  which  moves  so  that  the  sum  of  its  distances 
from  any  number  of  planes  is  constant. 

9.  Transform  the  equation  z"^  =  ax -{■  by  by  turning  the  axes  of  x  and  y 
around  the  «-axis  until  the  new  y-axis  coincides  with  the  line  ax  +  by  =  0,  z  =  0. 

Ans.  02  -  xVa^  +  52. 

10.  What  is  the  equation  of  the  surface 

x^  +  y^  -{-  2  z^  -  2  z(x  +  y)  =  a^ 

when  referred  to  new  axes  such  that  the  new  ic-axis  is  equally  inclined  to  OX, 
O  F,  and  OZ,  and  the  new  y-axis  is  the  line  x  +  y=0,  0  =  0?    Ans.  y^  +  Sz^  =  a^. 

11.  Show  that  the  six  planes,  each  passing  through  one  edge  of  a  tetrahedron 
and  bisecting  the  opposite  edge,  meet  in  a  point. 

12.  Through  the  middle  point  of  every  edge  of  a  tetrahedron  a  plane  is  drawn 
perpendicular  to  the  opposite  edge.  Show  that  the  six  planes  so  drawn  will  meet 
in  a  point  such  that  the  centroid  of  the  tetrahedron  is  midway  between  it  and 
the  centre  of  the  circumscribed  sphere. 

13.  Through  two  fixed  straight  lines  in  space  two  planes  are  drawn  at  right 
angles  to  one  another.    Find  the  locus  of  their  line  of  intersection.     (See  §  147.) 

14.  A  line  of  constant  length  has  its  extremities  on  two  given  straight  lines. 
Find  the  equation  of  the  surface  generated  by  it,  and  show  that  any  point  on  the 
line  describes  an  ellipse. 

16.  A  straight  line  meets  two  given  straight  lines  and  makes  the  same  angle 
with  both  of  them.    Find  the  equation  of  the  surface  which  it  generates. 

16.  Three  straight  lines  mutually  at  right  angles  meet  in  a  point  P,  and  two 
of  them  intersect  the  axes  of  x  and  y  respectively,  while  the  third  passes  through 
the  fixed  point  (0,  0,  c) .    Show  that  the  equation  of  the  locus  of  P  is 

x^  +  i/^  +  z^  =  2cz. 

17.  Show  that  when  the  new  axes  are  chosen,  as  in  Ex.  10,  the  equation  of  the 
surface  xy  +  yz  +  zx=:0  becomes  2x^  —  y^  —  z^  =  0. 


CHAPTER  XV 
CONICOIDS 

152.  A  surface  whose  equation  is  of  the  second  degree  is  called  a 
Conicoid.  In  this  chapter  we  shall  investigate  some  of  the  properties 
of  the  conicoids  by  taking  the  equations  of  these  surfaces  in  their 
Standard  Forms.  We  shall  begin  with  the  Sphere,  which  may  be 
defined  as  the  locus  of  a  point  whose  distance  from  a  fixed  point  is 
constant.  From  this  definition  it  follows  at  once  from  equation  (2), 
§  129,  that,  if  the  centre  is  at  the  origin  and  the  radius  is  r,  the 
equation  of  the  sphere  is 

x^-]-y^  +  z^  =  r^;  (1) 

and  if  the  centre  is  at  the  point  (a,  b,  c),  the  equation  is 

(X  -  a)2  +  (2/-  6)2  +  (;s  _  c)2  =  ^2,  (2) 

Moreover,  the  general  equation 

x^  +  y^  +  z^  +  2Ax-h2By  +  2Cz  +  D  =  0,  (3) 

may  be  written  in  the  form 

(x-^Ay  +  iy  +  Bf-hiz-^Cy^A'  +  B'+C'-D,  (4) 

which  shows  that  the  equation  represents  a  sphere  whose  centre  is 
the  point  (—A,  —Bj  —C)  and  whose  radius  is  -y/ A^ -{- B^ -\- C^  —  D. 
That  is,  every  equation  of  the  second  degree  in  which  the  coefficients 
of  a^,  2/^  and  z^  are  equal,  and  in  which  the  terms  containing  xy,  yz, 
and  zx  do  not  appear,  represents  a  sphere. 

153.  To  find  the  equation  of  the  tangent  plane  at  any  point  (a;',  y\  z') 
of  the  sphere. 

Let  the  equation  of  the  sphere  be  x^ -\- y"^ -\- z^  =  i^.  (1) 

The  equations  of  the  radius  drawn  to  {x\  y\  z')  are 

^=2^  =  ^.  (2) 

x'     y'     z'  ^^ 

Since  the  tangent  plane  passes  through  the  point  (x\  y\  z')  and  is 

perpendicular  to  this  radius,  its  equation  is 

220 


154]  CONICOIDS  221 

x'(x-x')-]-y'(i/-y')+z'{z-z')  =  0,  (3) 
Since  x'^  -\-y'^  +  z^  =  7^,  this  equation  reduces  to 

icx'  +  yy'  +  zz'  =  r^,  (4) 
In  like  manner,  the  equation  of  the  plane  tangent  to  the  sphere 

ix^-{-y^  +  z^-^2Ax  +  2By-^2Cz-\-D  =  0  (5) 
at  the  point  (a;',  y',  z')  can  be  shown  to  be 

xx'-\-yy'  +  zz'  +  A(x  +  x')+B(y  +  yf)  +  C(z  +  z')+n  =  0.  (6) 

154.  Interpretation  of  the  expression 

(x'  -  ay  +  (y'  -  by  +  (z'  -cy-cP  (1) 

when  the  point  P(x',  y\  z')  is  not  on  the  sphere 

(x^ay+(y-by-{-(z-cy-d'  =  0.  (2) 

Let  I,  m,  n  be  the  direction  cosines  of  any  line  through  P.  Then 
the  equations  of  this  line  may  be  written  (§  145) 

I  m  n  '  ^  ' 

Let  this  line  intersect  the  sphere  in  the  points  Q  and  B.  Then  at 
the  points  Q  and  E  [from  (2)  and  (3)] 

(lr  +  h'-ay'^(mr-\-y'-by-{-(nr-\-z'-cy-d^  =  0.  (4) 

If  Ti  and  ra  are  the  roots  of  this  equation,  we  have 

nrg  =  (X'  -  a)2  +  (y'  -  &)2  +  (2,/  _  c)2  -d^  =  rQ.  JPB,  (5) 

That  is,  the  expression  (1),  or  (5),  is  always  equal  to  the  product 
of  the  distances  from  P  to  the  sphere  measured  along  any  straight 
line  passing  through  P. 

If  7'i  Vz  is  negative,  P  is  inside  the  sphere.  Then  (5)  is  the  product 
of  the  segments  of  any  chord  passing  through  P;  it  is  also  numei-i- 
■cally  equal  to  the  square  of  the  radius  of  the  small  circle  on  the 
sphere,  whose  centre  is  at  P. 

If  ri  rj  is  positive,  P  is  outside  the  sphere.  In  this  case  the  expres- 
sion (5)  is  equal  to  the  product  of  the  whole  secant  by  the  external 
segment ;  and  therefore  it  is  also  equal  to  the  square  of  any  tangent 
PT  drawn  from  P  to  the  sphere,    (Cf  §  104.) 

Cor.  All  tangents  drawn  from  an  external  point  to  a  sphere  are 
equal. 


222  CONICOIDS  [155 

155.  If  a  sphere  passes  through  the  line  of  intersection  of  two  given 
spheres,  tangents  drawn  from  any  point  on  it  to  the  two  given  spheres 
are  in  a  constant  ratio. 

Let  S  =  x'  +  y'  +  z'  +  2Ax  +  2By  +  2Cz  +  D  =  0,  (1) 

and  S'  =  x'-}-y'-{-z'  +  2A'x-^2B'y-{-2C'z  +  D'  =  0,  (2) 

be  the  equations  of  two  spheres,  in  each  of  which  the  coefficient  of 
x^  is  unity.  Then  the  equation  of  any  sphere  through  their  line  of 
intersection  is  <^  __^^i-.q  (3) 

If  PTf  and  PT'  are  tangents  drawn  from  any  point  on  (3)  to  (1) 
and  (2)  respectively,  it  follows  from  §  154  that 

JPT2  =  X.1>T'2,  (4) 

which  proves  the  proposition,  since  X  is  constant  for  any  particular 
sphere. 
If  X  =  1,  equation  (3)  reduces  to 

2iA-A')x  +  2(iB-B')y  +  2(C-C')z  +  D-I>'  =  0,  (5) 

which  is  of  the  first  degree,  and  therefore  represents  a  plane. 

The  plane  through  the  line  of  intersection  of  two  spheres  is  called 
their  Radical  Plane.  The  radical  plane  of  two  spheres  may  also  be 
defined  as  the  locus  of  all  points  from  which  tangents  drawn  to  the 
two  spheres  are  equal. 

EXAMPLES 

1.  What  does  the  constant  term  D  represent  in  the  general  equation  of  the 
sphere  ?  Where  is  the  origin  if  D  is  positive  ?  if  Z)  is  zero  ?  if  Z>  is  negative  ? 
Where  is  P  in  §  154  if  nrz  =  -  fZ^  ? 

2.  How  many  independent  conditions  can  a  sphere  be  made  to  satisfy  ? 

8.   Find  the  equation  of  a  sphere  through  four  given  points. 

Find  the  centres,  radii,  position  of  the  origin,  length  of  tangents  from  the 
origin,  and  the  intercepts  of  the  following  spheres. 

4.  a;2  +  ?/2  +  2;2_2ic_42/-60  +  5  =  O. 

5.  x^  +  y^  +  z^  +  10x-2iy  =  0. 

6.  x:^-\-y^  +  z'^  +  6x-8y  +  2z-10  =  0. 

7.  a;2  +  y2  +  02  _  4  X  +  6  2/  +  10  5!  =  0. 


155]  CONICOIDS  223 

Find  the  equation  of  a  sphere 

8.  With  centre  on  one  of  the  coordinate  axes  and  passing  through  the  origin. 

9.  Touching  two  of  the  coordinate  planes. 

10.  Touching  the  three  coordinate  planes.  —  - 

11.  Touching  two  of  the  coordinate  axes. 

12.  Touching  the  three  coordinate  axes. 

13.  Touching  the  three  axes  and  passing  through  the  point  (2,  4,  0). 
How  many  such  spheres  are  there  ? 

14.  Show  that  if  the  coordinates  of  the  extremities  of  a  diameter  of  a  sphere 
are  (xi,  yi,  zi)  and  (x2,  Vi,  Zi)  its  equation  may  be  written 

ix  -  xi)  (x  -  Xi)  +  (y-  yi)  (y  -  2/2)  +  («  -  zi)  {z  -  Zi)  =  0. 

16.   Show  that  the  polar  equation  of  the  sphere 

x2  +  2/2  +  02^_2^4ic  +  2^y  +  2Cfe  +  2>  =  O 

is  p2  +  2  p{Al  +  5m  +  Cw)  +  D  =  0. 

What  property  of  the  sphere  follows  from  the  fact  that  the  product  of  the 
roots  of  this  last  equation  is  constant  ? 

16.  Show  that  the  radical  plane  of  two  spheres  is  perpendicular  to  their  line 
of  centres,  and  bisects  all  their  common  tangents. 

17.  Show  that  the  radical  planes  of  three  spheres  meet  in  a  line  which  is 
perpendicular  to  the  plane  through  the  centres  of  the  spheres. 

This  line  is  called  the  Radical  Axis  of  the  three  spheres. 

18.  Show  that  the  radical  planes  of  four  spheres  meet  in  a  point. 
This  point  is  called  the  Radical  Centre  of  the  four  spheres. 

19.  What  is  the  geometric  property  of  the  radical  axis  of  three  spheres  ?  of 
the  radical  centre  of  four  spheres  ?  What  is  the  analytic  condition  that  the 
origin  shall  be  the  radical  centre  of  four  spheres  ? 

20.  A  and  B  are  two  fixed  points,  and  P  a  variable  point  such  that 
PA  =  n  '  PB.  Show  that  the  locus  of  P  is  a  sphere.  Show  also  that  all  such 
spheres,  for  different  values  of  n,  have  a  common  radical  plane. 

21.  Show  that  the  spheres  whose  equations  are 

x*+2/2  +  «2  +  2^  +  2JBy  +  2(7«  +  2>  =  0 
and  a;2  +  ya  _^  2;2  ^  2  ^'a;  +  2  B'y  +  2  C'2?  +  2>'  =  0 

will  cut  one  another  at  right  angles  if 

2  ^14'  +  2  JB£'  +  2  CC"  -  2>  -  D'  =t  a 


224 


CONICOIDS 


[156 


The  Cone 

156.   To  find  the  equation  of  a  cone  generated  by  a  straight  line 
passing  through  the  origin,  of  which  the  guiding  curve  is  a  conic. 

Let  the  equations  of  the  guiding  conic  be 


«?^y' 


1,  z  =  c. 


(1) 


a'  '  b' 

Let  Q(xi,  i/i,  c)  be  any  point  on 
the  guiding  conic ;  then 


^4-^ 


1, 


(2) 


a-      b^ 

and  the  equations  of  the  generating 
line  OQ  are 

X  _y     z 
Xi~yi     c 
Whence 


(3) 


^1  =  -,   yi  =  -,    and 


-,    . =  1.     (4) 

r  cr  ^  ^ 

Substituting  these  values  in  equa- 
tion (2)  gives 

^    ^    f  ^  z^ 
a'r^'^b'r'     c'r' 


2  ^  62       c2 


0, 


(5) 
(6) 


which  is  the  required  equation. 
By  putting  x,  y,  and  z  respectively  equal  to  zero  in  (6),  we  find  the 
equations  of  the  traces  of  the  cone  to  be 


b'     (^     ^' 


^  +  2^  =  0. 


(7) 


Each  of  the  first  two  of  these  sections  is  a  pair  of  straight  lines 
through  the  origin,  and  the  third  is  a  point  ellipse. 

By  putting  cc,  y,  and  z  respectively  equal  to  fc,  we  find  the  equations 
of  the  three  sets  of  contours  to  be 


?!_^  — ^       ^  —  — 


x^     y 


k' 


¥     c^ 


(8) 


156]  CONICOIDS  225 

each  of  which  for  different  values  of  k  represents  a  system  of  similar 
and  coaxial  conies  (§  116).  The  first  two  are  hyperbolas  with  trans- 
verse axes  along  the  2J-axis,  and  whose  asymptotes  are  the  traces  on 
the  corresponding  coordinate  planes.  The  last  are  ellipses  which 
increase  indefinitely  in  size  as  the  cutting  plane  recedes  from  the 
origin.  As  a  check  it  should  be  noticed  that  the  section  made  by 
the  plane  2  =  c  is  the  guiding  conic. 
If  we  take  as  the  guiding  conic  the  hyperbola 

the  equation  of  the  surface  will  be 

which  is  a  cone  extending  along  the  2/-axis,  since  the  sections  per- 
pendicular to  this  axis  are  ellipses  with  centres  on  this  axis. 
Similarly,  if  the  guiding  conic  is  the  hyperbola 

-2-^2  =  1, 2;  =  c,  (11) 

the  resulting  equation  will  be 

052      1/2      «2 

which  represents  a  cone  extending  along  the  aj-axis. 
If  we  take  as  the  guiding  conic  the  parabola 

2/2  =  4aajj  z  =  c,  (13) 

the  equation  of  the  cone  will  be 

cy'^  =  4  axz,  (14) 

The  traces  of  this  surface  show  that  the  cone  is  tangent  to  the 
coordinate  planes  a;=0,  2=0,  along  the  2;-axis,  and  a^axis  respectively ; 
I.e.  these  axes  are  elements  of  the  cone.  The  a»-contours  are  the 
rectangular  hyperbolas  4  00:2;  =  cT^.  The  other  two  sets  of  contours 
are  the  parabolas       ^,  ^  ^  ^^^  ^^^  ^  ^  4  ^^^  ^^^^ 

Observe  that  these  parabolas  are  sections  made  by  planes  parallel  to 
an  element  of  the  cone. 


CONICOIDS  [157 

If  we  transform  equation  (14)  by  turning  the  axes  of  x  and  z 
clockwise  through  an  angle  of  45°,  the  new  equation  will  be 

^  +  |l_?!  =  0,  (16) 

c      2a     c  ^    ^ 

which  is  of  the  same  form  as  equation  (6),  and  therefore  represents  a 
cone  extending  along  the  new  2;-axis.  It  follows  from  equations  (6), 
(10),  (12),  and  (16)  that  the  conical  surface  generated  is  essentially 
the  same,  whatever  the  form  of  the  guiding  conic. 

The  equations  of  the  cone  found  above  are  all  homogeneous. 
Moreover,  if  they  are  referred  to  any  new  set  of  rectangular  axes 
having  the  same  origin  [(1),  §  151],  the  new  equations  will  also  be 
homogeneous.  Furthermore,  any  homogeneous  equation  represents  a 
cone  whose  vertex  is  at  the  origin.  For  if  the  coordinates  of  the  point 
(a;,  y,  z)  satisfy  a  homogeneous  equation,  so  also  will  the  coordinates 
of  the  point  (kx,  ky,  kz),  whatever  the  value  of  k  may  be.  Hence  a 
line  through  the  origin  and  any  point  on  the  surface  lies  wholly  on 
the  surface. 

157.  Definitions.  — The  form  of  equations  (6),  (10),  (12),  and  (16) 
of  §  156  shows  that  the  surfaces  which  they  represent  are  sym- 
metrical with  respect  to  each  of  the  coordinate  planes,  and  also  with 
respect  to  the  origin.  That  is,  each  of  these  planes  bisects  all  chords 
of  the  surface  which  are  perpendicular  to  the  plane.  A  plane  which 
bisects  all  chords  of  a  conicoid  which  are  perpendicular  to  it,  is 
called  a  Principal  Plane.  The  sections  made  by  the  principal  planes 
are  called  the  Principal  Sections  of  the  conicoid.  The  lines  of  inter- 
section of  the  principal  planes  are  called  the  Axes  of  the  conicoid; 
they  are  also  the  axes  of  the  principal  sections.  The  point  of  inter- 
section of  the  principal  planes  is  called  the  Centre  of  the  conicoid. 

It  follows  from  these  definitions  that  the  cones  in  §  156  have  three 
principal  planes  and  three  axes.  These  are  the  coordinate  planes 
and  the  coordinate  axes,  with  the  single  exception  of  the  locus  of 
equation  (14).  Moreover,  we  have  also  found  in  §  156  that  if  the 
guiding  conic  is  an  ellipse,  a  hyperbola,  or  a  parabola,  the  cone  is 
such  that  sections  perpendicular  to  one  of  its  axes  are  ellipses.  Such 
a  cone  is  called  an  Elliptic  Cone  to  distinguish  it  from  the  cone  of 
revolution,  or  circular  cone. 


158] 


CONICOIDS 


227 


The  Ellipsoid 


158.  Let 


+  -,  =  1,  y  =  0; 


and 


^  +  ^'  =  1,2  =  0, 
or     Ir 


(1) 

(2) 


be  two  fixed  ellipses,  XZ,  XY^  having  a  common  major  axis;  and 
let  ABC  be  a  variable  ellipse  which  moves  so  that  its  plane  is 


always  parallel  to  the  2/2;-plane,  and  which  changes  in  size  so  that 
the  ends  of  its  axes,  A  and  B,  always  lie  in  the  two  fixed  ellipses. 
The  surface  generated  by  this  variable  ellipse  is  called  an  Ellipsoid. 
Let  Pix,  y,  z)  be  any  point  in  the  ellipse  AB^  whose  semi-axes  are 
CA,  CB ;  and  let  PD  be  drawn  perpendicular  to  CA.  Then,  since 
DP  =  y  and  CD  =  Zy  yi         ^2 


CB^'^CA'       ' 


(3) 


Since  A  and  B  are  also  on  the  two  fixed  ellipses  (1)  and  (2), 
respectively,  and  their  coordinates  are  (x,  0,  CA)  and  (x,  CB,  0),  we 

have  ^  +  M'  =  l,  and  ^  +  «^ 


a* 


y" 


1. 


(4) 


228  CONICOIDS  [15d 


Substituting  in  (3)  the  values  of  Gj^  and  CI^  given  by  equa 
tions(4),weget  ^    ^    ^^ 


'^^-^'--=1,  (5) 


which  is  the  standard  equation  of  the  ellipsoid. 

The  surface  is  symmetrical  with  respect  to  each  of  the  coordinate 
planes,  and  also  with  respect  to  the  origin.  Hence  these  are  the 
principal  planes  of  the  surface,  the  coordinate  axes  are  its  axes,  and 
the  origin  is  the  centre. 

The  principal  sections  are  the  ellipses 


.+%  =  ^>     ^.  +  ^  =  1^     fe  +  :i  =  i-  (^) 


The  equations  of  the  three  sets  of  contours  are 

t^t^l-^       tj^t^l-t       t^t^l-^.         m 
y"     c"  a"'       a^     &  h^       a^     W-  (?  ^^ 

Each  set  is  a  system  of  similar  ellipses  which  vanish,  respectively, 
when  Tc  is  equal  to  ±  a,  ±  6,  ±  c. 

In  general,  it  is  here  assumed  that  a';>h'>  c. 
If  c  =  &,  the  equation  (5)  becomes 

The  2/2;-contours  are  now  concentric  circles,  and  the  surfaxje  is 
an  ellipsoid  of  revolution  generated  by  revolving  the  ellipse 
6V  H-  ay  =  a^W  about  its  major  axis.  This  surface  is  called  a 
Prolate  Spheroid. 

If  6  =  a,  the  equation  of  the  surface  (5)  takes  the  form 

05^       J/2       «2 

of  which  the  a;2/-contours  are  concentric  circles.  The  surface  is  an 
ellipsoid  of  revolution  generated  by  revolving  the  ellipse  (1)  about 
its  minor  axis,  i.e.  the  z-axis.  Such  a  surface  is  called  an  Oblate 
Spheroid. 

If  c  =  6  =  a,  the  ellipsoid  becomes  the  sphere 

a^  +  2/^  +  2^  =  a\  (10) 


159] 


CONICOIDS 


229 


159.   Let 

and 


The  Hyperboloid  of  One  Sheet 
a? 

,2         ^2 


T2-^  =  l^  2/  =  0; 


(1) 
(2) 


.^ 

'^ 

z     /• 

^^ 

V 

Hv    y 

/ 

^  ^ 

\ 

v"-""\ 

/ 

\ 

c 

/r 

D     ~'-^ 

V"^— I 

/ 

''  / 

/J*" 

\-""l 

---- 

J 

/ 

/ 

o 

\ 

/'"' 

Zt""-A 

/ 

^~— i 

/ 

HA 

/ 

-7 

—  -  —  i 

\ 

/"" 

/ 

z^-'-X 

'v^ 

// 

/ 

be  two  fixed  hyperbolas,  EFy  HK,  having  a  common  conjugate  axis 
OZ',  and  let  ABC  be  a  variable  ellipse  which  moves  so  that  its  plane 
is  always  parallel  to  the  an/-plane,  and  which  changes  its  size  so  that 
the  ends  of  its  axes,  A  and  JB,  always  lie  in  the  two  fixed  hyperbolas. 
The  surface  generated  by  this  variable  ellipse  is  called  a  Hyperboloid 
of  One  Sheet. 


230  CONICOIDS  [169 

Let  PiXy  ?/,  z)  be  any  point  in  the  ellipse  AB^  and  let  PD  be  drawn 
perpendicular  to  AC)  then,  since  CD  =  x,  DP  =  yj  and  CA,  CB  are 
the  semi-axes  of  the  ellipse, 

^      I       y      _  i  /Q\ 

CA^'^  CB""       '  ^^ 

Since  ^  and  B  are  on  the  fixed  hyperbolas  (1)  and  (2), 

which  is  the  standard  equation  of  the  hyperboloid  of  one  sheet. 

The  surface  is  symmetrical  with  respect  to  each  of  the  coordinate 
planes,  and  also  with  respect  to  the  origin.  Hence  the  coordinate 
axes  are  the  axes  of  the  surface,  and  the  origin  is  its  centre. 

The  principal  sections  made  by  the  planes  x  =  0  and  y  =  0  are  the 
two  fixed  hyperbolas  (2)  and  (1),  and  the  section  made  by  the  plane 
2  =  0  is  the  ellipse  ^     ^,2 

^  +  1  =  1-  (6) 

a^     Ir 

The  intercepts  on  the  axes  of  x  and  y  are  ±  a  a,nd  ±  6,  but  the 
surface  does  not  intersect  the  2-axis. 
The  equation  of  the  a^-contours  made  by  2;  =  A;  is 

-2  +  r2  =  i  +  l'-  (7) 

a^     W  (f 

These  sections  are  similar  coaxial  ellipses  for  all  values  of  A;,  which 
increase  in  size  without  limit  as  the  cutting  plane  recedes  in  either 
direction  from  the  origin. 

The  equation  of  the  contours  made  by  the  plane  a;  =  A;  is 

^-^=1-^-  (8) 

These  sections  are  hyperbolas  for  all  values  of  A;.  If  —  a  <  A;  <  a 
these  hyperbolas  have  their  transverse  axes  along  the  2/-axis,  but  if 
A;  >  a,  or  A:  <  —  a,  their  transverse  axes  lie  along  the  z-axis.  When 
A;  =  ±  a,  the  contour  is  a  pair  of  straight  lines,  which  are  the  asymp- 
totes of  the  entire  system  of  hyperbolas. 


160]  CONICOIDS  231 

Similarly,  the  contours  made  by  y  =  A;  are  the  hyperbolas 

which  have  their  transverse  axes  along  the  a>axis,  or  2-axis,  according 
as  A;<,  or  >6,  numerically,  and  whose  common  asymptotes  are  the 
contours  made  by  the  planes  y  =  ±b. 

When  b  =  a,  the  equation  (5)  of  the  surface  becomes 

^+IL*_5j  =  i,  (10) 

a2     a2     c2       '  ^    ^ 

which  is  a  hyperboloid  of  revolution  generated  by  revolving  the 
hyperbola  (1)  about  its  conjugate  axis. 

160.  Asymptotic  cone  of  the  hyperboloid  of  one  sheet. 
Let  the  equation  of  the  hyperboloid  be 

be  the  equation  of  a  cone  along  the  z-axis  [(6),  §  156]. 

The  equations  of  the  contours  of  these  two  surfaces  made  by  the 
plane  z  =  k  are,  respectively, 

^  +  2^-1  +  ^  (3) 

^a  y2  J^ 

and  a^  +  h  =  r  W 

A  comparison  of  equations  (3)  and  (4)  shows  that,  for  the  same 
finite  value  of  k,  the  section  of  the  cone  is  smaller  than  the  section 
of  the  hyperboloid.  Hence  the  cone  may  be  said  to  lie  inside  of  the 
hyperboloid. 

Equation  (3)  may  also  be  written  in  the  form 

which  shows  that  the  sections  of  the  two  surfaces  become  equal,  i.e. 
they  approach  the  same  limit,  when  the  cutting  plane  recedes  in 
either  direction  to  an  infinite  distance  from  the  origin.    That  is,  the 


282 


CONICOIDS 


[161 


cone  is  tangent  to  the  hyperboloid  at  infinity,  and  is,  therefore,  called 
the  Asymptotic  Cone  of  the  hyperboloid  of  one  sheet. 


161.  Let 


Thh  Hyperboloid  of  Two  Sheets 


and 


=  1,  2/  =  0; 


1,  .  =  0, 


(1) 


be  two  fixed  hyperbolas,  EF,  EHy  having  a  common  transverse 
axis;  and  let  ABC  be  a  variable  ellipse  which  moves  so  that  its 


plane  is  always  parallel  to  the  ysi-plane,  and  which  changes  its  size 
so  that  the  ends  of  its  axes,  A  and  B,  always  lie  in  the  two  fixed 
hyperbolas.  The  surface  generated  by  this  variable  ellipse  is  called 
a  Hyperboloid  of  Two  Sheets. 

Let  P{Xy  y,  z)  be  any  point  on  the  ellipse  AB,  and  let  PD  be 
drawn  perpendicular  to  CA\  then,  since  CD  =  Zy  DP  =  y,  and  CA, 
CB  are  the  semi-axes  of  the  ellipse, 


CB^'^CA'     ^' 
Since  A  and  B  are  also  on  the  fixed  hyperbolas  (1)  and  (2), 

-i ;;^  =  l,and- ^  =  1. 


a' 


a' 


b' 


(3) 


W 


161]  CONICOIDS  233 

which  is  the  standard  equation  of  the  hyperboloid  of  two  sheets. 

The  surface  is  symmetrical  with  respect  to  each  of  the  coordinate 
planes  and  the  origin.  Hence  the  axes  of  coordinates  are  the  axes, 
and  the  origin  is  the  centre  of  the  surface. 

The  intercepts  on  the  a^axis  are  ±  a,  but  the  surface  does  not 
intersect  either  of  the  other  axes. 

The  equation  of  the  contours  made  by  the  plane  a;  =  A;  is 

These  sections  are  imaginary  for  all  values  of  k  between  4-  a  and 
—  a.  Hence  there  are  no  real  points  on  the  surface  between  the 
planes  x  —  a  and  x—  —  a.  If  A:  is  numerically  greater  than  a,  these 
sections  are  real  ellipses  which  increase  indefinitely  in  size  as  k 
increases  without  limit,  but  reduce  to  points  when  k  =  ±a.  Hence 
the  planes  x  =  ±  a  are  tangent  to  the  surface.  Thus  the  surface 
is  shown  to  consist  of  two  distinct  parts,  and  for  this  reason  the 
hyperboloid  is  said  to  have  two  sheets. 

The  xy  and  a;2;-contours  are  hyperbolas  with  transverse  axes  along 
the  iP-axis,  and  whose  asymptotes  are  the  traces  of  the  asymptotic 
cone  on  the  xy  and  a;2j-planes.  From  §  160  it  is  evident  that  the 
equation  of  the  asymptotic  cone  is 

a2     62     c2     "•  ^^^ 

If  c  =  6,  equation  (5)  becomes 

a;2     y^     2:2 

^~62-^-l'  W 

which  is  the  equation  of  a  two-sheeted  hyperboloid  of  revolution 
generated  by  revolving  the  hyperbola  (2)  about  its  transverse  axes. 

Two  conicoids  are  similar  if  their  principal  sections  are  similar 
conies.     Hence,  if  K  is  an  arbitrary  parameter,  the  equations, 

represent  systems  of  similar  conicoids  (§  116). 


234 


CONICOIDS 


[162 


The  Elliptic  Paraboloid 

162.  Let  ABC  be  a  variable  ellipse  whose  plane  is  always  parallel 
to  the  a^-plane,  and  whose  vertices  A,  B  move  along  the  two  fixed 
parabolas  OA  and  OB,  whose  equations  are 

x'  =  2az,  2/  =  0;  (1) 

and  2/'  =  2&^,  a;  =  0.  (2) 

•The  surface  generated  by  this  moving  ellipse  is  called  the  Elliptic 

Paraboloid. 

Let  P(xj  y,  z)  be  any  point  in  the 
ellipse  AB,  and  let  PD  be  perpen- 
dicular to  AC ;  then  since  CD  =  x, 
SiJid  DP  =  y 


CA^^CB"       ' 


(3) 


Since  A  and  B  are  also  on  the  para- 
bolas (1)  and  (2),  respectively,  and 

OC=z 

CA^  =  2az, 

and  CB'  =  2bz.  (4) 


X 


y 


a      o  ' 


(5) 


which  is  the  standard  equation  of  the 
elliptic  paraboloid. 

The  surface  is  symmetrical  with 
respect  to  the  xz  and  yz  planes,  and  the  2;-axis.  Hence  the  2;-axis  is 
called  the  axis  of  the  paraboloid.  The  surface  passes  through  the 
origin,  cutting  the  z-axis  once,  the  x  and  y  axes  each  twice,  but  does 
not  cut  the  axes  at  any  other  point. 
If  we  put  z  =  km  (5),  we  get 


?  +  f  =  2^- 
a      0 


(6) 


Hence  a  section  parallel  to  the  aJ2/-plane  is  imaginary  if  k  is  negative. 
If  k  is  positive,  the  section  is  an  ellipse  which  increases  in  size  as 
the  plane  recedes  from  the  origin,  and  diminishes  to  a  point  when 
fc  =  0.  Therefore  the  surface  is  tangent  to  the  a^-plane,  and  lies 
wholly  above  this  plane. 


163] 


CONICOIDS 


235 


The  equations  of  the  xz  and  2/2;-contours  are 

a^  =  2az-^,ajidf  =  2bz-—' 
b  a 


(7) 


From  equations  (1)  and  (2)  we  see  that,  for  all  values  of  A;,  these 
sections  are  respectively  equal  to  the  two  fixed  parabolas  OA  and  OB. 
If  6  =  a,  equation  (5)  may  be  written 

ac2  +  2/2  =  2a«,  (8) 

which  represents  a  paraboloid  of  revolution  about  the  2;-axis. 

The  Hyperbolic  Paraboloid 

163.  Let  a^=:2az,  y  =  0,  (1) 

be  the  equations  of  a  fixed  parabola  OA,  and  let  AE  be  another 
given  parabola  with  a  constant 
latus  rectum  2  6.  Let  the  parab- 
ola AE  move,  keeping  its  vertex 
A  in  the  fixed  parabola  OA,  its 
plane  parallel  to  the  2/2;-plane,  and 
its  axis  AR  in  the  os^-plane,  the 
concavities  of  the  two  parabolas 
being  turned  in  opposite  directions. 
The  surface  generated  by  this 
moving  parabola  AE  is  called  a 
Hyperbolic  Paraboloid. 

Let  P{x,  y,  z)  be  any  point  on 
the  parabola  AE.  Draw  PD  per- 
pendicular to  AR ;  DC  and  AB  perpendicular  to  OZ. 

Then        BA^^x'^^^a  -  OB,  and  DP^  =  y''=^2h  •  DA. 

Whence       ^-^=  OB-DA  =  OC=z. 

2a     26 

a      b  ' 

which  is  the  standard  equation  of  the  hyperbolic  paraboloid. 

The  surface  is  symmetrical  with  respect  to  the  planes  x  =  0  and 
y  =  0,  and  the  z-axis.  Hence  the  2-axis  is  called  the  axis  of  the  sur- 
face. The  surface  cuts  the  z-axis  in  one  point,  the  x  and  y  axes 
each  in  two  coincident  points  at  the  origin. 


(2) 
(3) 
(4) 


236 


CONICOIDS 


[163 


If  we  put  z  =  km  equation  (4)  we  get 

a      b  ' 


(5) 


which  represents  a  hyperbola  with  transverse  axis  on  the  aj-axis  or 
y-axis  according  as  k  is  positive  or  negative.  When  A;  =  0,  the  section 
is  two  straight  lines,  HK  and  LM  (large  figure),  which  are  the 
asymptotes  of  all  these  contours. 


The  equations  of  the  xz  and  yz-contovLTs  are 

a^  =  2a2  +  ^,  and2/'  =  -26«  +  ^, 
b  a 


(6) 


which  for  all  possible  values  of  k  represent  two  systems  of  parabolas. 
The  first  are  all  equal  to  the  fixed  parabola  OA  with  axes  turned 
upward,  the  second  are  all  equal  to  the  movable  parabola  AE  with 
axes  turned  downward. 


164]  CONICOIDS  237 

164.  The  paraboloids  are  the  limiting  forms  of  the  central  conicoids 
as  the  centre  recedes  to  infinity. 

Let  the  equations  of  the  central  conicoids  be  _ 

If  the  origin  is  moved  to  the  point  (—  a,  0,  0),  the  new  equation 
may  be  written  ^     ^^     ^^2 

52  c* 

Let  —  =  l,  and  ~  =  l'  j  then  i,  V  are  respectively  the  semi-latera 

recta  of  the  principal  sections  made  by  the  planes  2  =  0,  and  i/  =  0. 
Equation  (2)  may  then  be  written 

aJ*     V^     25*     f»  /ON 

Now,  if  a  becomes  infinite,  while  I  and  V  remain  finite,  equation  (3) 
becomes  in  the  limit,  for  the  ellipsoid,  hyperboloid  of  two  sheets, 
and  one  sheet,  respectively, 

f+F=2»'    f+f=-2.,    t-t=2..  (4) 

The  first  two  are  elliptic  paraboloids,  the  last  is  a  hyperbolic  parabo- 
loid, all  with  axes  coinciding  with  the  aj-axis. 

EXAMPLES 

1.  Show  that  a  hyperboloid  degenerates  into  a  cone  when  its  axes  become 
indefinitely  small,  preserving  a  finite  ratio  to  eacb  other. 

2.  Show  that  the  traces  of  the  asymptotic  cone  are  the  asymptotes  of  the 
contours  of  the  hyperboloids. 

3.  Compare  the  section  of  the  hyperboloid  of  one  sheet  [(5),  §  159]  made 
by  the  plane  x  =  k  with  the  section  of  its  asymptotic  cone  made  by  the  plane 
X  =  y/k^  -  a*.    What  does  this  show  ? 

4.  Show  how  an  elliptic  paraboloid  may  be  generated  by  a  moving  parabola. 
6.   Show  how  a  hyperbolic  paraboloid  may  be  generated  by  a  moving  hyper- 
bola. 

6.  Show  that  all  planes  parallel  to  the  axis  of  a  paraboloid  cut  the  surface  in 
parabolas. 

7.  Show  that  the  projections,  on  a  plane  perpendicular  to  the  axis  of  a  para- 
boloid, of  all  plane  sections  not  parallel  to  the  axis,  are  similar  conies. 


238  CONICOIDS  [165 

8.  Show  that  all  parallel  parabolic  sections  of  a  paraboloid  are  equal. 

9.  Let  ri,  r2,  rg  be  any  three  semi-diameters  of  an  ellipsoid  which  are 
mutually  at  right  angles.     Show  that 

i.  +  ^  +  ±  =  l  +  l  +  l. 
n^     n^     rs^     a2     &2^c2 

10.  Show  that  the  equation  of  the  cone  whose  vertex  is  at  the  origin  and 
which  passes  through  all  the  points  of  intersection  of  the  ellipsoid  [(6),  §  158] 
and  the  plane  Ix  +  my  +  nz  =  1  is 

/v2        f<2        «2 

11.  Show  that  the  two  conjugate  hyperboloids 

have  a  common  asymptotic  cone,  and  show  how  they  are  situated  with  respect 
to  this  cone. 

12.  What  are  the  limiting  forms  of  the  asymptotic  cones  as  the  hyperboloids 
pass  into  paraboloids  in  §  164  ? 

Tangent  Planes 

165.    To  find  the  equation  of  the  tangent  plane  at  any  point  {x\  y\  z') 
on  a  conicoid. 
Let  the  equation  of  the  conicoid  be 


Let  the  equations  of  any  line  through  the  point  (x',  y\  2')  be 

I  m  n  \         /     \  / 

or  x  =  x^  +  lry    y  =  y'-{-mr,    z  =  z'-\-nr.  (3) 

The  distances  from  the  point  (ic',  y'^  z')  to  the  points  where  this  line 
meets  the  conicoid  are  the  values  of  r  given  by  the  equation 

(x'  +  lry     (y'^mrf     (z'  +  nry_ 

a'       +        P        "^        ?       -^'  ^^^ 

JP  ,  m2     n2\      „    fix'  .  my'  .  nz'\  .  x'^  ,  y''  ,  z"     ^      ^     .^, 


Since  the  point  (x',  y',  z')  is  on  the  conicoid, 

x^     y^.z^ 
^2  +  52  +  ^2 


„12        ,/2        «f2 


165]  CONICOIDS  239 

Therefore  one  value  of  r  is  zero,  whatever  the  direction  of  the  line 
(2)  may  be.  But  if  we  choose  the  direction  of  the  line  so  that  we 
also  have  7  ,  ,         , 

^2+  52  +  ^  -"»  — -iJ; 

the  other  value  of  r  will  also  vanish ;  that  is,  the  line  will  then  meet 
the  surface  in  two  coincident  points,  and  is  therefore  a  tangent  line 
at  the  point  (a;',  y\  z'). 

The  equation  of  the  locus  of  all  the  tangent  lines  which  can  he  drawn 
through  the  point  {x\  y\  2')  is  found  by  eliminating  Z,  m,  n  between 
equations  (2)  and  (7).     We  thus  obtain 

^(»-«')+^(2'-/)  +  5(^ -«')  =  <>,  (8) 

which,  by  virtue  of  equation  (6),  reduces  to 

XX'     yy'     zz'  _  .^. 

Hence  the  tangent  lines  all  lie  in  a  plane.  This  plane  is  called  the 
Tangent  Plane  at  the  point  (a;',  y\  z'). 

By  a  proper  choice  of  signs  in  (9)  we  can  write  the  equation  of  the 
tangent  plane  to  either  of  the  hyperboloids. 

It  should  be  noticed  that  the  factors  before  the  parentheses  in 
equation  (8)  can  be  obtained  by  taking  half  the  partial  derivatives 
(§  61)  of  equation  (1)  with  respect  to  x,  y,  z,  respectively,  and  then 
substituting  in  these  derivatives  a;'  for  a;,  y'  for  y,  and  2'  for  %.  It 
can  be  shown  that  this  rule  holds  for  any  surface. 

Assuming  this  rule  to  hold  for  the  paraboloids 

J±f-2.  =  0,  (10) 

we  have  for  the  tangent  plane  at  the  point  (a;',  y\  z') 

J(a5-aj')±|'(2/-y)-(^-«')=0,  (11) 

or  ^^yf=^^  +  z').  (12) 

This  should  also  be  proved  independently. 

Ex.  Show  by  means  of  equation  (5)  that  every  plane  section  of  a  conicoid  is 
a  conic. 


240  CONICOIDS  [166 

166.  The  Normal  to  a  surface  at  any  point  P  is  the  straight  line 
through  P  perpendicular  to  the  tangent  plane  at  P. 

Hence  the  equations  of  the  normal  to  the  ellipsoid  at  the  point 
(«',  y\  z')  are  [(9),  §  165] 

-  {irJ       1M  —  «#'«  —  «' 

(i) 


a2 

y  -y'  z-z' 
~     y'     ~    z'    ^ 

and  to  the 

elliptic 

paraboloid  [(11),  §  165] 

y  -y'  z-z' 
-    y         -1' 

(2) 
a  b 

From  these,  by  a  proper  choice  of  signs  in  the  denominators,  we 
easily  obtain  the  normals  to  the  other  conicoids. 

167.    To  find  the  condition  that  the  plane 

Ix -\- my -\- nz  =  p  (1) 

shall  touch  the  ellipsoid. 

The  equation  of  the  tangent  plane  at  any  point  {x\  y\  z')  of  an 
ellipsoid  is  [(9),  §  165] 

Equations  (1)  and  (2)  will  represent  the  same  plane  if 

p       p      p  a^       ly"       c^        '  ^  ^ 


Equating  the  coefficients  of  the  identity  (3)  gives 

l_x^     'rfi_y[     ^_^' 
p     a^    p      H^    p~^ 


(4) 


Whence  a^l±^^ri_±^^x-     y^     z^^^^ 

p"  a^^  V"^  &  ^^ 

Therefore  the  plane  Ix -\- my -\- nz  =  p  will  touch  the  ellipsoid  if 

a2«2  +  62^2  +  ^2^2  =p2,  (6) 

In  like  manner  it  can  be  shown  that  the  same  plane  (1)  will  touch 
the  paraboloid  ^       • 

-4-^  =  2«  (7) 

ah 

if  al^  +  hm^  +  2  pn  =  0.  (8) 


160]  CONICOIDS  241 

168.  To  find  the  locus  of  the  point  of  intersection  of  three  tangent 
planes  to  an  ellipsoid  which  are  mutually  at  right  angles. 

Let  the  equations  of  the  three  tangent  planes  be  [(6),  §  167] 

?!»  +  wiiy  +  ni«  =  VaV  4- ft^w^i^  +  c^V>  —  (1) 

l^c  4-  rn^  -\-n^=  ^o^li  +  h'^rti}  +  (?n},  (2) 

and  l^  -h  may  +  7132;  =  VaV  +  6^3^  +  <^n}.  (3) 

Squaring  and  adding  these  equations  we  get,  by  virtue  of  the 
relations  between  the  direction  cosines  of  mutually  perpendicular 
lines  (§  151),  aj2  +  y2  +  2;2  =  a2  +  62  +  c2.  (4) 

Therefore  the  required  locus  is  a  sphere.  This  sphere  is  called 
the  Director  Sphere  of  the  ellipsoid. 

Poles  and  Polar  Planes 

169.  The  equation  of  the  plane  tangent  to  the  conicoid 

t^tj^t^X  (1) 

at  the  point  (a;',  y\  2'),  if  this  point  is  on  the  surfa^e^  is  (§  165) 

a^  h^  cf 
Suppose,  however,  that  the  point  {x\  y\  2*)  is  not  on  the  surface. 
What,  then,  is  the  meaning  of  this  equation  (2)  ?  It  still  represents 
a  real  plane,  which  is  related  in  some  definite  way  to  the  point 
(x\  y'f  z^  and  to  the  conicoid,  since  its  parameters  involve  both  the 
coordinates  of  the  point  and  the  parameters  of  the  conicoid.  In 
order  to  determine  what  this  relation  is,  we  will  let 

x  =  x'  +  lr,  y  =  y'  +  mr,  z  =  z'  +  nr  [(3),  §  165]  (3) 
be  the  equations  of  any  straight  line  through  the  point  (a;',  y',  2'). 
Substituting  these  values  of  a;,  y,  z  in  equation  (2),  we  find  the  dis- 
tance from  the  point  («',  y',  2')  to  the  point  where  this  line  meets  the 
plane  (2)  to  be  the  value  of  r  given  by  the  equation 

te|  ,  my^  ,  nz' 

r~       (r^^       7/'*       «'2  •  \*) 


-  +  ^-4- 
a*  ^  6»  ^  c 


242  CONICOIDS  [169 

Let  Vi  and  r2  be  the  distances  from  the  point  (x*,  y\  z')  to  the  points 
where  this  line  (3)  meets  the  conicoid  (1).    Then  from  equation  (5), 

§  165,  we  get 

Ix^     my[  ,  n£ 

2      11  2ri^2  fa\ 

.'.  -  =  —  +  —  ,  or    r  = =-=-•  (6) 

r    ri     r^  ^i  +  rg 

That  is,  the  plane  (2)  and  the  point  (x\  y\  z*)  divide  harmonically 
every  chord  of  the  conicoid  (1)  drawn  through  the  point  (x',  y\  z'). 

This  plane  is  called  the  Polar  Plane  of  the  point  («',  y'  z'),  and  the 
point  (x\  y\  z')  is  called  the  Pole  of  the  plane  with  respect  to  the 
conicoid.     (Cf.  §  94.) 

If  ri  =  rj,  the  line  (3)  is  tangent  to  the  surface.  But  when  Vi  =  9*2, 
we  find  from  equation  (6)  that  r  =  ri  =  rg. 

Therefore  the  polar  plane  passes  through  the  points  of  contact  of  all 
tangent  lines  drawn  from  its  pole  to  the  surface. 

The  assemblage  of  such  tangent  lines  forms  a  cone,  which  is  called 
the  Tangent  Cone  from  the  point  to  the  surface. 

Moreover,  if  ri  =  0  and  ?'2  =^  0,  then  r  =  0  also,  in  whatever  direc- 
tion the  line  is  drawn ;  i.e.  if  the  point  («',  y',  z')  is  on  the  conicoid, 
it  is  also  on  its  own  polar  plane.  If  ri  =  r2  =  0,  then  r  is  indetermi- 
nate ;  i.e.  when  the  line  is  tangent  to  the  conicoid  it  lies  wholly  in 
the  plane. 

Therefore  the  pole  of  a  tangent  plane  is  the  point  of  contact. 

When  the  point  (x',  y\  z')  coincides  with  the  centre  of  the  conicoid, 
ri  =  —  r2,  and  therefore  ?•  =  oo. 

Hence  the  polar  plane  of  the  centre  is  at  infinity. 

Furthermore,  the  second  of  equations  (6)  shows  that  r  is  always 
realf  although  ri  and  ra  may  be  imaginary.  This  is  evidently 
necessary,  since  the  line  will  always  meet  the  plane  in  one  real 
point. 

In  a  similar  manner  it  can  be  shown  that  equation  (12),  §  165, 
is  the  polar  plane  of  the  point  (x',  y\  z')  with  respect  to  the  parabo- 
loids. 


170]  CONICOIDS  243 

170.  If  the  polar  plane  of  a  point  P,  with  respect  to  a  conicoid,  passes 
through  a  point  Q,  then  will  the  polar  plane  of  Q  pass  through  P. 

The  proof  of  this  proposition  is  precisely  the  same  as  that  of  the 
corresponding  proposition  in  Plane  Geometry  (§  95).  -_  _ 

Let  R  and  S  be  any  two  points  on  the  line  of  intersection  of  two 
planes  A  and  B,  whose  poles  with  respect  to  the  same  conicoid  are 
P  and  Q.  Then,  since  E  is  on  both  of  the  planes  A  and  B,  the  polar 
plane  of  R  will  pass  through  both  P  and  Q,  and  therefore  through 
the  line  PQ.  For  the  same  reason  the  polar  plane  of  S  will  pass 
through  the  line  PQ.  Similarly,  the  polar  plane  of  any  point  Pj  on 
the  line  PQ  will  pass  through  the  line  RS. 

The  two  lines  PQ  and  RS  which  are  such  that  the  polar  plane, 
with  respect  to  a  conicoid  of  any  point  on  the  one,  passes  through 
the  other,  are  called  Polar,  or  Conjugate  Lines. 

EXAMPLES   ON   CHAPTER   XV 

1.  Show  that  every  tangent  plane  to  a  cone,  and  the  polaj  plane  of  any  point 
(except  the  vertex)  with  respect  to  a  cone,  passes  through  the  vertex. 

2.  Show  that  all  normals  to  a  sphere  pass  through  its  centre. 

3.  Show  that  the  line  OP  joining  the  centre  O  of  a  sphere  to  a  point  P  is 
perpendicular  to  the  polar  plane  of  P.  If  the  line  OP  meets  the  polar  plane  in 
Q,  show  that  OP'OQ  =  r^. 

4.  Show  that  the  distances  of  two  points  from  the  centre  of  a  sphere  are 
proportional  to  the  distances  of  each  from  the  polar  plane  of  the  other. 

6.  Show  that  the  locus  of  the  point  of  intersection  of  three  mutually 
perpendicular  tangent  planes  to  a  paraboloid  is  a  plane. 

6.  Find  the  equation  of  the  director  sphere  of  the  surface  generated  by 
revolving  a  rectangular  hyperbola  around  its  conjugate  axis. 

7.  Show  that  tangent  planes  at  the  ends  of  a  diameter  of  a  conicoid  are 
parallel. 

8.  Prove  that  the  locus  of  the  poles  of  a  series  of  parallel  planes  is  a  straight 
line  through  the  centre  of  the  conicoid. 

9.  Find  the  equation  of  a  sphere  which  cuts  four  given  spheres  orthogonally. 
[See  Ex.  21,  p.  223.] 

10.  Show  that  a  sphere  which  cuts  each  of  the  two  spheres  S=0  and  S'  =  0 
at  right  angles,  will  also  cut  the  sphere  S  +  \S'  =  0  a.t  right  angles. 


244  CONICOIDS  [170 

11.  Find  the  equation  of  the  sphere  which  touches  the  plane  y  =  0,  and  cuts 
the  plane  2:  =  0  in  the  circle  (x  —  ay  +  (y  —  by  =  r^.  Show  that  the  area  of 
the  section  of  the  sphere  made  by  the  plane  x  =  0  is  ir{b^  —  a^).  Why  is  this 
result  independent  of  r  ? 

12.  A  straight  line  is  drawn  through  a  fixed  point  0,  meeting  a  fixed  plane 
in  Q,  and  in  this  line  a  point  P  is  taken  such  that  OP-  OQ  ia  constant.  Show 
that  the  locus  of  P  is  a  sphere  passing  through  0,  whose  centre  is  on  the  line 
through  O  perpendicular  to  the  plane. 

13.  A  straight  line  moves  so  that  three  fixed  points,  A,  B,  C,  on  the  line  lie 
one  in  each  coordinate  plane.  Show  that  any  other  point  P  on  the  line  generates 
an  ellipsoid  whose  semi-axes  are  equal  to  PA,  PB,  and  PC. 

14.  Show  that  the  equation  of  the  cone  whose  vertex  is  at  the  centre  of 
the  ellipsoid,  and  which  goes  through  all  points  common  to  the  ellipsoid  and  the 
sphere  x^ -{■  y^ -\-  z"^  =  r%  is 

16.  If  a  >  6  >  c  and  r  =  6  in  Ex.  14,  show  that  the  cone  breaks  up  into  two 
planes,  whose  intersections  with  the  ellipsoid  are  circles. 

16.  If  P  and  Q  are  any  two  points  on  an  ellipsoid,  the  plane  through  the 
centre  and  the  line  of  intersection  of  the  tangent  planes  at  P  and  Q  will  bisect 
the  chord  PQ. 

17.  P  and  Q  are  any  two  points  on  an  ellipsoid,  and  planes  through  the 
centre  parallel  to  the  tangent  planes  at  P  and  Q  cut  the  chord  PQ  in  P'  and  Q'. 
Show  that  PP'  =  QQ'. 

18.  The  normal  at  any  point  P  of  an  ellipsoid  meets  a  principal  plane  in  G. 
Show  that  the  locus  of  the  middle  point  of  PG  is  an  ellipsoid. 

19.  The  normal  at  any  point  P  of  an  ellipsoid  meets  the  principal  planes  in 
d,  ^2»  Cfs.     Show  that  PG^  PG2,  PGz  are  in  a  constant  ratio. 

20.  The  normals  to  an  ellipsoid  at  the  points  P,  P'  meet  a  principal  plane  in 
G,  G'.     Show  that  the  plane  which  bisects  PP'  at  right  angles  bisects  GG'. 

21.  Show  that  a  section  of  a  hyperboloid  made  by  a  plane  parallel  to  an 
element  of  the  asymptotic  cone  is  a  parabola. 

22.  Show  that  the  general  equation  of  a  cone  referred  to  three  of  its  generators 
as  axes  of  coordinates  is  fyz  +  gzx  +  hxy  =  0. 


APPENDIX 

I.     The  Direction  of  a  Curve  at  the  Origin. 

It  is  often  useful  to  know  how  to  find  the  direction  of  a  curve  at 
the  origin  before  taking  up  the  formal  study  of  slope.  In  many 
instances  this  can  easily  be  done. 

For  example,  let  the  equation  of  the 

curve  be  „  ... 

2/  =  ar'.  (1) 

Let  P{x,  y)  be  any  point  on  the  curve 
close  to  the  origin.  Draw  the  line 
OPj  and  let  B  represent  the  angle 
XOP. 

Then  tan^==^=^. 


OD     X 
Since  the  point  P  is  on  the  curve,  we  have,  from  equation  (1), 


tan^: 


(2) 


(3) 


The  direction  of  the  curve  at  the  origin  is  the  limiting  direction 
of  the  line  OP  as  we  make  the  point  P  move  along  the  curve  and 
approach  as  near  as  we  please  to  the  origin.  From  equation  (3)  we 
get  for  this  limiting  direction  of  OP 


J™   tan  «  =  ,«-„(.) 


0. 


W 


That  is,  the  direction  of  the  curve  at  the  origin  is  the  same  as  the 
direction  of  the  x-axis. 

If  the  equation  of  the  given  curve  is 


then 


2/  =  a^-ar,  or  -  =  a^-l, 
245 


(5) 
(6) 


246  APPENDIX 

Hence  the  direction  of  the  curve  at  the  origin  is  that  of  the  line 
y  =  -x.     (See  the  curve  FQ  in  §  27.) 

The  direction  of  the  curve  at  the  origin  can  be  found  in  this  way 
whenever  the  equation  of  the  curve  can  be  put  in  the  form 

i  =  *(»').  (7) 

provided  we  can  find  the  limiting  value  of  <;^  (x)  as  a;  =  0. 

Moreover,  the  direction  of  a  curve  at  the  points  where  it  crosses 
the  axes  can  be  found  in  a  similar  manner.  For  example,  the  locus 
of  equation  (5)  cuts  the  fl>axis  at  the  point  (1,  0).  Let  this  point  be 
R,  and  let  P(x,  y)  be  a  point  on  the  curve  close  to  B  such  that 
x>l.    Let  $  be  the  angle  XEP. 

Then  tan  (9  = -J- =  ^  4- a;.  (8) 

X—  1 

.■.^f^t^ne  =  ^%(^  +  x)=2.  (9) 

Hence  the  curve  has  the  direction  of  the  line  y  =  2(x—  1). 


EXAMPLES 
Find  the  direction  of  the  following  curves  at  the  origin  : 

I.  y  =  x^.  2.   y  =  X".  3.   y'^  =  x^. 

4.   y'^  =  ax.  5.   x^-y\a-x)  =  0.  6.   a;(x2  +  ?/2)- a(x2  -  y2)  =  0. 

Find  the  direction  of  the  following  curves  at  the  points  where  they  cut 
the  axes : 

7.   ?/  =  x3-3a;2  +  2x.     (See  Ex.  2,  §  81.)  8.  2/ =  a;^  -  x^. 

9.   y  =  x8-x2-6x.  10.    ?/  =  x3-2x2-llx  +  12. 

II.  Example  illustrating  §  81. 

Let  f'(x)  =  2kx.  (1) 

Then  f(x)=kx'  +  c,        '  (2) 

where  c  is  an  arbitrary  constant  which  will  disappear  when  we  take 
the  derivative. 


APPENDIX 

Then  y  =  2'kx=f{x) 

is  the  equation  of  the  straight  line  L'M\  and 

y  =  kx'^-c  =  f{x) 
is  the  equation  of  the  parabola  LGM^  where  OG  =  c. 

lY  My 


247 
(3) 


Let  OQ  =  a  and  OB  =  h.  Then  QA'  =  2  fca,  RB'  =  2  fc6,  the  area 
of  the  triangle  OQA'  —  ka^^  and  the  area  of  triangle  OBB'  =  kb\ 

.'.  area  of  QRB'A'  =  kV  -  ka\  (5) 

Also,  RB=f(b)=^kh''^c,  and  qA=f{a)=^ko?^-c  [from  (4)].  (6) 

.-.  BB-QA  =  f(b)  -  /(a)  =  fc62  _  fcal  (7) 

.-.  area  of  QRB'A'  =f(b)  -/(a)  =  RB  -  QA.  (8) 

That  is,  the  number  of  square  units  in  the  area  of  the  trapezoid  is 
equal  to  the  number  of  linear  units  in  (RB—  QA). 

Similarly,  the  area  of  EFD'C  =  EC—  FD,  a  negative  number. 

If  we  put  c  =  0,  the  parabola  will  pass  through  the  origin,  and  the 
ordinate  QA  will  be  zero  when  the  area  of  the  triangle  OQA'  is  zero. 
Then  the  number  of  units  in  the  ordinate  QA  will  be  equal  to  the 
number  of  units  in  the  area  of  the  triangle  OQA'. 


248  APPENDIX 

III.    Trigonometrical  formulce. 

1.  sin  ^  CSC  ^  =  1.  8.   sin  (—  ^)  =  —  sin  6. 

2.  cos  ^  sec  ^  =  1.  9.  Cos  (—  6)  =  cos  0. 

3.  tan  ^  cot  ^  =  1.  10.   sin  (90°  ±  $)=  cos  6. 

4.  tan^  =  ^.  11.   cos(90°±^)=Tsin^. 

5.  sin2  $  4-  cos^  6  =  1.  12.   sin  (180°  ±  ^)  =  =F  sin  0. 

6.  sec2  6  -  tan^  ^  =  1.  13.   cos  (180°  ±6)  =-  cos  $. 

7.  csc^  ^  -  cot^  ^  =  1.  14.   sin  (270°±^)  =  -cos^. 
16.  cos  (270°  ±^)=  ±  sin  9. 

16.  sin  (6  ±  6')  =  sin  6  cos  0'  ±  cos  6  sin  $'. 

17.  cos  (9  ±  9')  =  cos  9  cos  9'  T  sin  9  sin  ^'. 

18.  tan(^±g^)=  tan^±tan^'^  ^^    ^^^2^=    ^^^^^ 


iTtan^tan^'  1-tan?^ 

20.   cot  (9  ±  9')  ^^ot^^ot^-Tl^  21^   ,ot 2 ^  =  22^iz-_l. 

^  ^       cot^'±cot^  2cot^ 

22.  sin2^  =  2sin^cos^. 

23.  cos2^  =  cos2^-sin2^  =  2cos2d-l  =  l-2sin2^. 

24.  sini-^  =  VKl-cos^).  25.   cos  J  ^  =  VJ(lTcos^. 

26.  sin  ^  +  sin  ^'  =  2  sin  ^(9  +  9*)  cos  |(^  -  9'). 

27.  sin  ^  -  sin  ^'  =  2  cos  ^(9  +  ^')  sin  i(9  -  9^ 

28.  cos  ^  4-  cos  0'  =  2  cos  ^(9  +  ^')  cos  ^(9  -  9'). 

29.  cos^-cos^'  =  -2sini(d  +  ^')sin|(^-^'). 

In  any  plane  triangle 

30     ^^^  ^  _  sin  ^  _  sin  C  gi     a  +  ^  _  tan -|(.^  4- ^) 

a  b  G    '  '    a  —  b     tani(^  — J5)' 

32.   a^  =  b^-\-c^  — 2  be  cos  A.  33.   Area  = -^  6c  sin  ^. 


ANSWEKS 


CHAPTER  I 

Page  4.-7.  (x,  -y),  (-«,  y),  (-«,  -y).  8.  (a\/2,  0),  (0,  aV2), 
(-  aV'2,  0),  (0,  -  ay/2).  9.  (0,  0),  (2a,  0),  (a,  aVS)  for  one  position  of  the 
triangle.  10.  On  a  line  two  units  to  the  right  of  the  x-axis.  On  a  line  three 
units  below  the  y-axis.  11.  Yes.  Yes.  No.  13.  y  =  x,  or  y  =  —  x.  14.  On  a 
Circle  with  centre  at  the  origin  and  radius  equal  to  a  \  x^  -^  y'^  =■ 'k.  15.  x=  —  5, 
or  X  =  1.    16.   y  =  X  +  3. 

Pages. —6.  An  isosceles  triangle.  7.  A  parallelogram.  8.  (VlO,  tan-i-^), 
(-  \/2,  45°),  ( VIO,  tan-i  -  3),  (3v^,  45°).  9.  (0,  0),  (2  a,  0),  (2  a,  60°)  ; 

(a,  0),  (a\/3,  30°),  (a,  60°).  10.   (0,  0),  (2  a,  0),  (2aV2,  45°),  (2  a,  90°)  ; 

(a,  0),  (aVB,  tan-4),  (aV5,  tan-i2),  (a,  90°).  11.    (0,  0),  (2a,  0), 

(2aV3,  30°),  (4  a,  60°),  (2aV3,  90°),  (2  a,  120°);   (a,  0),  (aV7,  tan-i^V 

(aVl3,  tan-i^V    (a\/l3,   tan-i2\/3),    (avT,  tan-i-§^V   (a,   120°). 

12.  p  must  vary  from  0  to  co,  while  d  varies  from  0  to  2  ir,  or  ^  must  vary  from 
—  00  to  4-  CO,  while  d  varies  from  0  to  tt.  19.  p  —  a  sec  ^,  where  a  is  the  distance 
from  the  pole  to  the  line ;  p  =  hc&cd.  20.  On  a  circle  passing  through  the  pole, 
with  centre  on  the  initial  line  and  diameter  a ;  on  a  circle  with  diameter  a,  above 
the  initial  line  and  tangent  to  it  at  the  pole. 

Page  9.— 1.   13.    3.   2\/7.    4.   3V5,  3V6,  3V2. 
Page  10.  — 3.  a^b-2y/2' 

Page  11.  - 1.    (-  2,  1)  and  (~  1,  2),  (4,  7)  and  (-7,-4).  2.  2  :  3  ; 

-(3:8). 

Page  16.  — 1.  13.  2.  ^VS.  3.  12^  +  6a/3.  4.  2V3.  6.  8\/3. 
6.  ia2V3.  7.  2ac.  8.  a\  9.  31.  10.  47.  11.  (2V3,  2),  (\/2,  -  V2), 
(!i-fV2).  12.  (5,tan-i-f),  (13,tan-i-J^),  (\/l0,tan-i3).  13.  (-^,0), 
(I,  1).  14.  (-  J^,  J^),  (-  \,  \),  16.  (17,  1),  (-  13,  6).  16.  Sides  13,  13, 
7\/2.  Medians  i  V366,  \y/m>,  ^y/2,  isosceles.  17.  Sides  4\/2,  3V2,  6V2, 
area  12,  a  rt.  A.  18.   Sides  2V5,  2V6,  3V5,  3\/5,  area  24,  a  parallelogram. 

19.  p  =  a.  20.  e=a.  21.  p  =  2asin^tan^.  22.  p  =  asecd  ±b.  23.  y=mx. 
24.  x2-2/2  =  flf2.    26.   (x2  +  ya)8  =  4a2xV.    26.   (2xH22/Hax)2=  a2(x«+y2). 

249 


250  ANSWERS 

CHAPTER  II 

Page  19.  —  1.  x^  +  2/2  >^  <^  or  =  9.  2.  (x  +  3)2  +  (y  -  1)2  >,  <,  or  =16. 
3.   (X  -  ay  +  {y-  6)2  >,  <,  or  =  r^. 

Page  20.  —  1.   ?/  -  X  +  2  >,  <,  or  =  0.     2.   y  +  ic  -  3  >,  <,  or  =  0. 
Page  21. —1.   3x  +  5?/-4>,  <,  or  =0. 

2.  2(rt  -  c)x  +  2(&  -  d)y  =  a"^  +  b^  -  c^  -  cP. 

Page  22.  —  1.  The  origin.  All  the  plane  except  the  origin.  No  locus  in 
the  plane.  2.  The  jc-axis.  No  locus.  All  the  plane  except  the  x-axis.  3.  The 
line  X  =  a.  All  the  plane  to  the  right  of  the  line  x  =  a.  All  the  plane  to  the 
left  of  the  line  x  =  a.  4.  The  line  y  =  b.  All  the  plane  above  this  line.  All 
the  plane  below  this  line.  5.  The  circular  ring  bounded  by  the  circles  x^+y^=4t 
and  x2  +  2/2  =  9.  6.   The  ring  bounded  by  the  concentric  circles  (x  —  2)2  + 

(y-  3)2  =  9  and  (x  -  2)2-f  (y  -  3)2  =  16.  7.   All  the  plane  between  the  two 

lines  x=  a  and  x  =  b.  8.  A  circle.  All  the  plane  outside  of  this  circle.  All 
the  plane  inside  of  this  circle.  9.  That  part  of  the  plane  bounded  by  two  circles 
passing  through  the  pole,  with  centres  on  the  initial  line  and  diameters  a  and  b. 
10.    Similar  to  No.  8.      11.    Similar  to  Nos.  8  and  10.    12.  x=2,  x=|,  x=3,  x=|. 

Page  33,  §  25.  — 1.  n.  2.  The  values  of  p  corresponding  to  ^  =  0  are  the 
intercepts  of  the  locus  on  the  initial  line.  The  values  of  6  corresponding  to 
p  =  0  give  the  direction  of  the  lines  tangent  to  the  curve  at  the  pole. 

Page  33,  §  26.  —  The  points  of  intersection  are  :   1.  (f|,  V)-     2.  (-^^,  V)- 

3.  (-2,-3).  4.  (0,6),  (3,  4).  6.  (8,1).  6.  Imaginary.  7.  (0,0), 
(-6,-2).  8.  (^av^,av^).  9.  la(2 -\-V5),  2a^2  ±Vll  10.  (4a,4a). 
13.  6|. 

Page  37.  —  The  loci  of  these  equations  are  symmetrical  with  respect  to : 
1.  2/-axis.  2.  j^-axis.  3.  x-axis.  4.  x-axis.  6.  Origin.  6.  Origin.  7.  x-axis. 
8.  2/-axis.  9.  Both  axes,  and  the  origin.  10.  Origin.  11,  12,  13,  14,  15, 
16.  Both  axes,  and  the  origin.  17.  Origin.         18.  Nothing.  19.  ?/-axis. 

20.  Nothing.  21.  The  origin,  and  the  lines  y  =  x  and  y  =  —  x.  22.  y-axis. 
23.  Origin.  24.  Origin.  25.  The  origin,  and  the  lines  y  =  x  and  ?/  =  —  x. 
26,  27.  The  line  y  =  x.  28.  The  origin,  both  axes,  and  both  lines  y  =  x,  y=z  —  x. 
29.  Both  axes,  and  origin.  30.  Origin.  31.  Same  as  28.  32.  The  line  y  =  x. 
33.  Same  as  28.  34.  x-axis.  35.  x-axis.  36.  y-Sixis.  37.  Origin.  38.  Both 
axes,  and  origin.    39.  The  origin,  and  the  lines  y  =  x,  y  =  —  x. 

Page  41,  §  32.  —  1.   x2  +  y^  =_±  2ry.  2.  a,  b,  r.  3.  (T  2,  0),  2. 

4.  (0,  T  3),  3.  5.  (-  1,  2),  V5.  6.  (|,  -  f),  ^Vsi.  7.  (-  3,  2),  2. 
8.  (I,  -1),3.    9.  (-3,  -4),  6.    10.  (f,4),4.    11.  x'^+y^±2rx±2ry+r^=0. 

Page  41,  §  33.  —  1.  (2),  (3),  (4)  are  of  the  first  degree  because  the  pole  is 
on  the  circle.  Hence  any  line  through  the  pole  can  cut  the  circle  in  only  one 
other  point.  In  (1)  the  pole  is  outside  if  a  >  r,  inside  if  a  <  r.  2.  See  for- 
mulae in  §  32. 


ANSWERS  251 

Page  43.  —  1.  p^  =  — r-^ ; Since  the  denominator  is  the  sum 

6^  cos2  e  +  a'^  sin2  d 

of  two  squares,  it  can  never  be  zero.  Hence  p  can  never  be  infinite.  2.  Out- 
side, inside.    4.    (1)  10,6,  (±4,0).     (2)  4>/2,4,  (±2,0).     (3)  10,8,(0,  ±3). 

Page  44,  1 36.  -3.  .-  ,.,„,. /T,. 3,„. ,-    Infinite.       4.   (1)  (±  5,  0), 

3x  +  4?/  =  0,  3a;-4?/  =  0._  (2)  (0,  ±\/4l),  25 2/2-16 a;2=0.  (3)  (±2>/6,  0), 
4x2-2/2  =  0.  (4)  (±V2,  0),  x2-?/2  =  o.  (5)  (0,  ±\/2),  x^  -  y^  =  0. 
(6)  (±V6,0),  4x2 -2/2  =  0. 

Page  44,  §  37. -(2)  (1,  0),  x  =  -  1.      (3)   (-  2,  0),  x  =  2.      (4)  (f,  0), 
2x  =  -3.     (5)  (0,2),  2/ =-2.     (6)  (0,  -  f),  2  2/ =  5.     (7)  (0,  -  3),  2/ =  3. 
Page  45.  — 1.  2/  =  4x.  3.  2(a  -  c)x  +  2(6  -  d)y  =  a2  +  52  _  ^2  _  ^. 

5.  x2  +  2/^  +  6x-62/  +  9  =  0.  6.  x2  +  2/2  ±  8x  ±  8?/ +  16  =  0.  7.  (x-2)2  + 
(2/  +  6)2  =  (4  ±  2)2.  11.    (1),  (2),  (4)  a  straight  line  ;   (3)  a  hyperbola. 

12.  (1)  a  circle  ;    (2)  a  hyperbola  ;   (3)  two  hyperbolas  ;   (4)  two  straight  lines. 

13.  x2  +  2/2  =  c2  -  a2.  14.  x2  +  2/2-4x  =  0.  15.  2ax  =  c2.  16.  A  circle  with 
centre  at  the  centre  of  the  square. 

CHAPTER  in 

Page  51.  — 1.   When  the  line  goes  through  the  origin.    When  the  line  is 

n  1  X    XI-  .        «  1  10        X  V  -    X      VSv 

parallel  to  the  2/-axis.    2.   y  = -—x -—, —^^  +  -^=1,  -  +  -^=  -  5 

V2 

3.  2/-fx  +  6,    3g  +  |  =  l,     -|a:  +  t2/  =  ¥.       4.   y  =  x-6,    ^  +  "^^  =  1 

V2      \/2  .  4     ,5  V41         V41         \/41 

6.  2/  =  Ax-l|,   ^^  +  _^  =  l,,5^x-H2/  =  l.     7.   2/  =  ix  +  |,    1^  +  1=1 

^-A.y  = 9_.        8.   y=ix-i,    ?  +  -iL=i,    .2_x-_3_'^  =  ^_, 

V6      V5  2V6  *        ^     2^-t       '    Vi3         VI3^      Vl3 

»•   2/  =  -|a;,    4^  +  4^  =  0.     10.    ?  =  1,   x  =  a.     11.   2/ =  4,    ?  =  1,   y  =  4 
\/l3      Vl3  a  4 

12.  ^^       :e+        ^^       2/+         ^^       =0;i^x  +  ^2/+^^  =  0 
^  +  Jg+0    ^  A  +  B+0^^  A  +  B+C        '    A  A^^       A^ 

J2A{A  +  B+C)^        I2B{A+B  +  C)        j2C(^  +  ^+C)  _ 
\  BG  ^y  CA  ^^y  AB 

13.  _iOa;-82/  +  40  =  0;  -^x-^2/  +  l  =  0.  14.  |x-32/  +  9  =  0; 
±  fa;  1=22/ ±6  =  0.  16.  -6x  +  22/±3  =  0,  or  20x  -  82/ -  12  =  0; 
10x-4  2/-6  =  0,  or  -  15x  +  62/  +  9  =  0. 

Page  52.  —  1.   a  =  30°,  p  =  2.     2.    a  =  60°,  ;)  =  1.    8.   «  =  -  45°,  p  =  3. 

4.  a= -120°,  j)= -4.        6.    «  =  120°,  p  =  -  1.        6.    «  =  -  60°,  i?  =  -  5. 


252  ANSWERS 

7.  pcoa{e  —  a)=p.  8.  pcoB0  =Pt  psine—p.  9.  ^  =  A;,  where  A;  is  any  con- 
stant angle  ;  ^  =  0.  10.  ^  =  0  ;  <?  =  0,  and  ^  =  90°  ;  ^  =  0,  ^  =  60°,  and  6  =  120°  ; 
n  straight  lines  through  the  pole  and  equally  inclined  to  each  other.  _  11.  Simi- 
lar to  Ex.  10.    12.    (0,  V2) .    13.^  =  0,  and  p  cos  (^  -  45°)  =  2  (  V6  -  V2) . 

Page  54. —1.   3a;  -  15y  +  10  =  0.    2.   x  +  y=±6y/2,    3.   a;  -  yV3  =  8. 

4.  a;vi  +  3 y  +  9  =  0.  6.  y  =  2x  -  10.  6.  ax  -  by -\- a^ -h  h^  =  0.  1.  x-^y 
4-10  =  0,  10 a; -1-7  2/ =  11,  (ia -2h)x- by  -  a^ -\- b"^ -\- 2 ab  =  0.  8.  y  +  ^x 
=  7,  x  +  2y  +  T  =  0,y  =  Sx.  9.  «=  1,  5x -}- 3y  =  0,  2x  -  3?/ =  7.  10.(1) 
a;-|-y  =  3,  (2    x  -  y  +  5  =  0.     11.   2z  +  Sy  +  l=:0. 

Page  57.  — 1.   45°.       2.   90°.      3.   45°.      4.   tan-if.      5.  tan'i  ^^^=-^. 

2ab 
6.  tan-i3|.  7.  4x  -  Sj/ +  1  =  0,  3x -|- 4?/=  18.  8.  y=  2x  -  10,  x -i- 2?/ =0. 
9.  3x -4y -1- 18  =  0,  4x  + 32^  =  1.  10.  10x-^y  =  S.  11.  25x-M5y  +  3 
=  0,  25x-M5y-f37  =  0;  5x -f  3y  =  81,  5x -f  3y -f  89  =  0.  12.  3x-|-4y 
=  20.  13.  (oo,od).  14.  4x-7y  =  5,  x-j-6y=13,  6x-22/  =  18.  15.  8x 
-14y=7,    X  4-5?/ 4-5  =  0,    lOx  -  4?/ 4- 3  =  0.  16.  tan-i  ||,    tan-iff, 

tan-i--^.    17.   y  =  (16il7\/3)x. 

Page  60.  — 1.   12i,  -7,7.      2.    -  VlO,  -  \/iO,  VIO.       3.    -f|,  -  i%, 

_^.     4. 4a&_,-j(:£+^^    (a-b)\     5.    i^,  _  31,  _  9i.    6.    7x 

Va2  -F  62        Va2  4-  6-2     VoMTp 
4-y  =  0,  x~7i/  =  24.      7.   llx-3i/4- 15  =  0,  21x4- 77y  4-6  =  0.     8.   x-y 
=  1,  x4-2/  +  13  =  0.     9.   x(l-V3)4-yCl+V^)=12,  x(l4-V3)-2/(l -V3) 

=  6.     10.    ^'» -^ , -^ .      11.   The  two  straight  lines  2x-4y  =  9±  6 VS. 
y/6   V58    V79 

Page  61.  — 1.  2x4-3y  =  0,  6x-5y  =  14,  8x4-5y  =  7.  2.  9x-4y 
=  23.  3.  9x4- 18y  4- 15  =  0,  y  =  2x.  4.  38x  -  19y  4- 2  =  0,  76x  -  57y 
=  444.  6.  x2  4-2/2_iOx  =  0.  7.  (1)  x^  4- y2  _  4^  4- 4y  =  0,  (2)x24-y2 
-8x4-8y-f  16  =  0. 

Page  63.  —  1.  (1)  x  =  0  and  x  4-  «y  =  0.  (2)  x  =  0,  x  4-  y  =  0,  x  -  y  =  0. 
(3)  X  4-  2/  =  0.  (4)  X  -  y  =  0.  (5)  ax+by  =  0,  ax-by  =  0.  (6)  The  origin. 
(7)  X  4-  1  =  0,  X  -  1  =  0,  y  4-  2  =  0,  y  -  2  =  0.  (8)  ax  4-  6y  4-  c  =  0,  ax+by 
-c  =  0.    (9)y4-«-a=0,y-x4-a  =  0.     (10)  The  point  (a,  6).     (11)  x  4- 2/ 

—  a  —  6  =  0,  X  —  y  —  a4-&  =  0.  (12)  x  —  y.  (13)  x  cos  a  4-  y  sin  a  =  a.  2.  2  * 
-62/4-7  =  0,  4x4- 52/ =  20.     3.   x  4- 32/ =  0,  4x  4- 32/ =  0.     4.   ax-by  =  a^ 

-  62.  6.  X  4-  2/  +  1  =  0,  2/  =  «  -f  3.  8.  f ,  ±  V3.  9.  p  cos  (^  -\-  30°)  =  ±3, 
xV3-2/  =  ±6.  10.   3x-22/  =  3.  11.   2|.  12.    (0,  3t^V2), 

(-21±20V2,0).      13.   2x4-ll2/  =  5.      14.     ^^  ~  ^^  .      16.   10x-62/  =  9. 

•     VZ2  4-  wi2 
21.  5x-32/  =  4,  3x-|-52/  =  16,  5x-32^4-13  =  0,  3x-f52/4-l  =  0.     22.  2/  = 
(_  2  ±  \/3)x  4-  75  q=  V3,  2/  =  (-  2  ±  \/3)x  4- 1  ±  V3.     25.  21||.     28.    {a)  7  x 
.-7  2/4- 40=0  or  7x4- 7 2/ =  32,    (6)  2x4- 2/ =  4.  29.   3x- 42/ 4- 25  =  0. 


ANSWERS  253 

30.  ic(80±\/95)-y(15=F2V95)  +  120  =  0.  31.  a;  =  3,  12a;  +  35y+ 104  =  0. 
32.  7a;  -  y  =  46,  5x  -  5y  =  88.  33.  x^ -^  y^  -h  2x  -8y  -^^  I  =0,  x^  -\-  y^  -\-  60x 
-200y  + 625  =  0. 

CHAPTER  IV 

Page  69.  —  l.y^  =  ix.  2.  2x^-\-Sy^=zl.  3.  x^-^y^  =  r^.  4.  xy  =  a^. 
5.   2/2  +  4aa;  =  0.     6.   x^-y^  =  0,     7.    ^  +  |^=1.     8.  2a;y  +  a2  =  0.     9.   x^ 

+  2y2  =  16.  10.  X  -  y\  11.  (a  +  K)x'^  +  (a  -  h)y'^  =  1.  12.  2y2  = 
a{2xV2--a).  13.    (a;^  +  y2)8  _  cf2(a;2  _  y2)2^   P  =  asin2^,    /)  =  acos2^. 

14.  xy/a^  +  b-^  =  ah.    15.  xy  =  ±  a\    16.  2a;2  +  82^2  _  i.    13.  45°.    19.  tan-i  \, 

ortan-i-2.       20.   tan-i  f ,  or  tan-i  ~  f.       21.   ^tan-i-?A_.      23.    The  two 

a  —  h 
curves  will  be  the  same  if  the  unit  of  the  scale  of  the  second  is  taken  twice  as 
large  as  the  unit  of  the  scale  of  the  first. 

CHAPTER  V 
Page  71.  — 1.   fa.     2.   2a2.     3.   0.     4.   f.     5.   2  a.     6.   2.     7.   ^     8.   0. 
Page  75.-1.   0,  8,  12,  ....         2.   0,  ±8,  ±32,  ....         3.  00,  T2,  Ti,  -. 

4.  0,  ±f,  ±fV2,  ....  6.  -4,-1,  8,  ....  6.  0,  =F36,  ^48,  ....  7.  -J, 
- 1, 00 ,  -  1,  -  i,  -  i,  -  T^^,  ...,  for  X  negative,  -  \,  ^'^,  ....  8.  0,  ±  i>/2, 
±1.     9.   0,  00 ,  ±  |V3.     10.    -  .3,  -  .1,  -  .6,  .5,  -  1,  1,  .... 

Page  78.  — 1.  3a;  +  4y  =  25.  2.  12a;  -  5y  +  169  =  0.  3.  2^  =  a;  +  2, 
2y  =  a;  +  8.  4.  2a;  +  2/  =  6.  5.  y  =  8a;-16,  a;  +  y  =  2.  6.  a;  +  3y  =  4. 
7.  2a5-3y  =  0,  2x  +  3y+18  =  0.  8.  3y  =  4a;-8,  4x  +  32/  =  26, 

4a;  +  3y  +  24  =  0,  3y  =  4a;4-42.  9.  x-^2y  =  S,  2y  =  x  +  5,  2y  =  x-5, 
x  +  2y  =  lS.  10.  a;  =  3,  3a;  +  4y  =  15,  4y-3a;  +  17.  17.  a;a;' -  yy' =  1. 
20.  a;a;'~-i  +  yy'^-^  =  1.  21.  a  =  1 ;  y  =  1.  22.  a;  +  2  y  =  3.  23.  y  =  2  a;  -  8. 
24.   2y  =  x  +  9,  «  +  22/  +  l=0.  25.   3!/=4a;-l,  4a;  +  3y  =  13. 

26.  a;  —  2/  =  0,  a;  +  2/  =  0,  a;  =  ±  a.  27.   The  equation  of  the  tangent  at  any 

point  (a,  /3)  on  the  curve  is  ^^^  +  ^^  =  2,  at  the  point  (a,  6),  -  +  ^  =  2. 

a«  &»*  a     b 

30.   tan-if.     32.  90°.     33.  90°.    34.  tan-i3^3j.    35.  2  y  =  a;  +  4,  2x  +  y  =  12, 

46°.        39.    -^^.      41.   -r^^^'        42.    -J^^ll-.        63.    -sin  2  a;. 
(1  -  a;2)2  (a  -  bxy  (1  +  a;)"+» 

55.  tan2  x.       56.   8  sec*  x.       57.   x  cos  x.     59.   2  sec2  x  tan  x(sec2  x  +  tan^  x). 

61.   4x8  +  2x(a  +  6).  62.   15x*  +  3x(2a  +  6x).  63.      ~  ^  "'^  . 

a  -  3  X  2  X  -4-  3  x8  ^^  "^  ^^ 

65-        ,  67.        /         '      68.   6x(2x«  +  3)(l-3x2)2(2x-12x8-9). 

2Va-x  Vl+x2  V        -T-    yv  j\  j 

^  71.    ^,f~^^'      73.   _i^i±J_.      74.  (m  sin2  X  +  n  cos2  X) 


(i  +  x2)t        a+«^^)* 

(8in**-^x  ■  cos"*-^x\ 
cogm+ix     sin»+ix  ] 


254  ANSWERS 


CHAPTER  VI 
Page  86.  —  1.  0(ic)  =  2x^  -5x^-6,  B  =  ^.      2.  2,  4.     3.  4,  14.     4.    ±  V2. 

5.  2±V3.  6.  '^^Yo~^^'  '^'  1,-3,4.  8.  ±1,3,5.  9.  Two  real  roots, 
one  between  0  and  1,  the  other  between  4  and  5.  10.  2,  —  4,  10.  11.  3,  and 
one  root  between  0  and  1,  the  other  between  —  1  and  —  2.  12.  4,  and  one  root 
between  0  and  1,  the  other  between  —  2  and  —  3.  13.  —  3,  5,  between  0  and  1, 
and  between  3  and  4.  14.  x^  +  x^  -  17  x +  15  =  0.  15.  x*  -  3  cc^  -  28  x"^ 
+  36  a; +144  =  0.  16.  30^3  +  77  x2  -  92x +  21  =  0.  17.  x*  -  17  a;2  +  16  =  0. 
18.  x*  -  2 x3  -  19x2  +  20  X  =  0.  19.  x*  -  5 x2  +  6  =  0.  20.  x*  +  2 x^  +  2  x2 
+  4  X  =  0.  21.  x3  -  13  x2  +  50  X  -  60  =  0.  22.  x*  -  6  x^  - 8x2  -  66x  -  65  =  0. 
23.  x5-3x*-23x3  +  61x2  +  94x-120  =  0.  24.  x^  +  2  x*  -  16x3  +  18x2 
+  15x  =  0.  25.  6x6 -11x5 -10  x*  + 3x3 +  2x2  =  0.  26.  x*  +  2  x^  +  9x2 
+  2x  +  66  =  0.    27.   x5  +  5x4-20x2  +  71x  +  231  =  0. 

Page  87.  —  1.-2.      2.    -  V3.       3.    -  4.      4.   4.      5.   0,  0,  0,  3  +  V^^. 

6.  0,  -  f .    7.   0,  0,  -  |. 

Page  90.— 1.  1.879,  -.347,  -1.532.  2.  1.356,1.692,-3.048.  3.  1.939. 
4.  3.264.  5.  1.769,2.672,  -4.441.  6.  .593,2.047.  7.  2.382,4.618.  8.  3.128, 
1.201,-1.33.    9.    .494,2.861,-3.112.     10.    2.583,7.169,-3.399,-6.353. 

Page  92.        2.  3,   -5,  2  -  y/3.       3.  f ,  3  -  V^.        7.  x*  +  2  x2  +  25  =  0. 

8.  ±V2±V3I,3±V363. 

4 
Page  95.-1.  x2  +  4x-5  =  0.        2.  x3-6x2- 7  x  +  60  =  0.       3.  x3  +  8x2 
-  28  X  -  80  =  0.  4.  X*  -  12  x2  -  12  X  +  3  =  0.  5.  x^  -  4  x2  -  60  x  =  0. 

6.  x*  -  4  x3  +  x2+  6  X  =  0.         7.  x*  -  11  x3  +  36  x2  -  30  X  =  0.        8.  x2  -  25  =  0. 

9.  a;3  _  4  a;  _  2  =  0.  10.  x*  -  35  x2  -  90  x  +  304  =  0.  11.  (1)  c  =  -  2,  (2)  c  =  1, 
or  -  5.  12.  x3  +  6  x2  -  32  =  0,  or  x3  -  6  x2  =  0.  13.  9  x2  +  8  x  -  1  =  0. 
14.  12x3  + 13x2-3x -2=0.     15.  24  x4  +  20x3 -30x2-5x  +  6  =  0.     19.  1,2, 

4.     20.-1,11.     21.   -1,  -1±2V^.     22.  i,l±2^.     23.  1,^I±^. 

o  5  4 

25.   ±1,  1±^-Z^.     26.  -2,  -I,  2  +  V3.       27.  -1,^,2, 


^i±V^n5.      28.  ±1,4.2,  ^i±|^E«. 
14  5 

Page  97.-1.   (2,  -16),  (-2,  16).        2.  (1,  16),  (4,  -11).        3.  (0,  32), 

(4,  0).  4.  (-1,  -67),  (3,  189).  5.  (0,  10),  (2,  -54),  (-2,  74). 

6.  (1,73),  (-3,  -567),  (4,  -224). 

Page  100.  —  1.  -  6,  -  6,  1.  2.  3,  3,  -  4.  3.  3,  3,  3,  -  2.  4.  2,  2,  3,  4. 
6.3,3,3,-4.  6.  2,  2,  2,  -  3,  -  3.  7.-1,-1,-1,2,2.  8.3,3,-2,-2. 
9.  1,  1,  1,   -3  +  V-15,      ^Q    2,  2,  2,  -3.      11.  1,  1,  1,  1.  -4.      12.    -4,-4, 


ANSWERS  255 

-4,2^        13.  _5,  -6, -5,  -5,1.         14.2,2,2,-1,-1,1.        16.  1  ±  v^ 

1±  V2,  -  2. ^16.   -  2  ±  V3,  -  2  ±  VS,  -  2,  3.      17.  1  ±  V^,  1  ±  V^,  -  4. 

18.  _l±V-2, -l±V-2,  4.       19.  4g8  +  27r2  =  0. 

Page  104.  —  1.  4,  -  4,  0.       2.   -  36,  18.       3.  108.        6.  34^2^.       6.  ^^  a'^. 

7.  f|V2.      8.  I.       10.  1.      11.  |.       12.  9a2.      15.  i- 

Page  111.  —1.  Max.  4  ;  min.  0  ;  (1,  2).     2,  Max.  4  ;  min.  -28 ;  (3,  -12). 

8.  Neither  a  max.  nor  a  min.;  (1, 11).     4.  Max.  36;  min.  32;  (3,  34).     6.  rV2; 
rV2  and  lrV2.        6.  aV2  and  ^6V2.        7.  f  the  altitude  of  the  segment. 

8.  The  square  with  corners  at  the  middle  points  of  the  sides  of  the  given  square. 

9.  ^  the  altitude  of  the  cone.    10.  h=r.    11.  h=2r.     12.  /i  =  fr.     13.  2x8x8. 

14.  l(a  -h  h—  \^a'^  —  ab  -\-  b'^),  where  a,  b  are  the  sides  of  the  given  rectangle. 

15.  He  must  walk  one  mile.         16.  6  miles  an  hour.         17.  |  rVS.         18.  |  r. 

19.  fr.      20.fr.     21.   V2.      22.  r=J-^,  h  =  J^.     23.  4  r.      24.  i  the 
altitude  of  the  paraboloid.     27.  10.392  in.  and  14.697  in. 


CHAPTER  VII 

Page  117.  — 5.  The  two  foci  of  a  circle  coincide  at  the  centre,  and  the  direc- 
trices are  at  an  infinite  distance  from  the  centre  ;  one  focus  and  one  directrix  of 
a  parabola  are  in  the  infinite  region  of  the  plane  ;  in  the  case  of  two  intersecting 
lines  the  two  foci  coincide  with  the  point  of  intersection  of  the  lines,  and  the 
two  directrices  also  coincide  and  pass  through  this  same  point. 

Page  130.  —1.  2x  +  y  +  2  =  0,  2y  =  a;  +  6.  2.  x-^y-{-2  =  0,  y  =  x-6. 
8.  ix-3y  =  25,  Sx-{-iy  =  0.  4.  6a;  +  3?/  +  16  =  0,  Sx-6y-\-30  =  0. 
5.  «-2?/  +  4  =  0,  2x  +  y  +  S  =  0.  6.  x  +  2y  +  2  =  0,  2x-2^  =  6.  7.  x  =  0. 
8.  2 ic  +  3 ?/  =  0.  d.  5x-Sy  =  0.  10.  2x-y  =  0.  11.  Put  the  first  degree 
terms  of  the  equation  of  the  conic  equal  to  zero  ;  the  result  will  be  the  equation 
of  the  tangent  at  the  origin.  12.   2y  =  Sx  +  6.         13.    x  +  2y-\-2  =  0. 

14.  2  a;  +  3  ?/  =  1.  15.  2  x  -  y  +  7  =  0.  16.  The  equation  of  the  polar  of  the 
origin  with  respect  to  a  conic  is  found  by  putting  half  of  the  first  degree  terms 
plus  the  constant  term  of  the  equation  of  the  conic  equal  to  zero.  17.  y  =  x  +  l, 
2y  =  x  +  4.  18.  6y  =  5x  +  9,  x  +  2y-f-5  =  0.  19.  3y  =  4x  +  25,  3x  +  42/  =  25. 
20.   2^  +  3  =  0,    4  X  +  5  2/  =  25.  24.   The  line  at  infinity.      At  infinity. 

25^    /r^cos«    r!lllL^V(-Psec«,    -tana). 
\     P  P      J 

CHAPTER  VIII 

Page  131.  —  3.  It  passes  through  the  point  (x  ,  y')  and  is  perpendicular  to 
the  polar  of  this  point. 


256  ANSWERS 

Page  133.  •—  1.  Tangents,  y  =x-\-a,  y  =  —  x  —  a;  normals,  y  =  -  a;  +  3 a, 
y  =  x-Sa.  3.  2/  =  \/3(a;  +  l).  ^.2y  =  2x  +  3.  5.  F(- 2,  0),  i^C- f,  0), 
L.R.  =  3,  directrix  a;  =  -  2|.  6.  F(-  2,  2),  F^-  2,  f),  L.R.  =  2,  directrix 
y  =  21.  7.  F(-  2,  4),  .F(-  i,  4),  L.R.  =  6,  directrix  x  =  3 J.  8.  F(3,  -1), 
j?'(3,  _ ^1),  L.R.  =  I,  directrix  y  =-  1^^.  9.  F(l,  3),  F{-  1,  3),  L.R.  =  8, 
directrix  x  =  S. 

Page  136.  — 1.  y=x-l.    2.  3ic-4y  +  l=0.    3.  2y  =  x  +  6,Sx+2y  +  2  =  0. 

a  r  3  ,-7 

6.  (1)    (a,    ±2  a),   (2)   (0,   0)  and  (3  a,  2aVB).      7.   2^  +  W?  +  «a/- =  0- 

'0         'a 

24.  (1)  y  =  A^a:,  (2)  kx/^  —  y^  +  2ax  =  0,  (3)  A;x  =  a,  where  A;  is  the  constant. 
26.  y'^  =  ax,  where  y^  _  4  Qj/g  is  the  given  parabola.  27.  y^  =  a(2x—a). 
28.  y'^-2ax'-ky  +  2ah  =  0,  (1)  1/2  =  2  a(a:  -  a),  (2)  y2  =  2ax,  (3)  2/2  = 
2  a(x  -  4  a),  (4)  y*  =  2  a(x  +  a). 

CHAPTER  IX 

Page  142. —1.  2(x^  +  y^)-Ux-ny  +  12  =  0.  2.  x2  +  y^-ix-\-2y  =  0. 
3.  ar2  +  y2  +  8 a;  +  20 2/  +  31  =  0.  4.  x^  +  y^ -ISx  -  y -i-lO  =  0.  5.  2 a;  +  3 y 
=  ±2Vl3.  6.  3a;-22/  =  9±3\/l3.  't.x  +  2y  =  5.  8.  9a; -6?/ =  14.  9.  6« 
-42/  =  14.    10.   (4,2),  (2,-6).    11.   (2,-4),  (3,  |).    12.    (-3,2). 

Page  143.  —  1.  The  point  (x',  y')  is  then  inside  the  circle.  2.  The  product 
of  the  segments  of  any  chord  (or  secant)  drawn  through  the  origin.  3.  The  ori- 
gin is  outside,  on,  or  inside  the  circle  according  as  c  is  positive,  zero,  or  negative. 

Page  146.  — The  products  of  the  segments  of  the  chords  in  examples  1  to  6 
are :    1.  9,  37.    2.    -  12,  7.     3.   15,  -  4.    4.   -  IJ,  7J.  .  5.  4,  25.    6.  - 11,  94. 

7.  5a; -6y +  4  =  0.  8.  16x  +  8y  =  17.  9.  h{a-h)y  =  ac.  10.  x-y  =  0, 
VJ^(a  +  6)2-4c.  12.  (0,1).  13.  (0,0).  14.  y  =  m(a;  -  a)  +  6,  where  (a,  6) 
is  the  centre.  15.  One,  viz.  the  line  through  the  given  point  and  the  centre. 
16.  a;2+?/2-6a;-4?/=0.  17.  x'^+y'^-Qx-^y+^-O.  18.  x^+y^-\-Qx-Qy-\-^-0, 
a;2+?/2+30a;-30y-|-225=0.  19.  4(a;2+y2^cx_ci/)  +  c2=0.  20.  2x+5y=9. 
2L  a;  +  5y  +  13  =  0.     23.  9  x  +  12  y  =  29,  9  a;  +  12  y  +  71  =  0.     25.(2,-3). 

26.  (1)  X  +  my  =  ±  r vTTwi2,     (2)   /x±yVc2-r2  =  cr,     (3)   x±y=±rV^. 

27.  14  X  -  12  y  +  29  =  0.  28.  ax  -  &y  =  a2.  29.  (f  f ,  ^j).  30.  (  -  2  a,  2  6). 
34.  ci  =  C2  =  cs,  where  ci,  cs,  cz  are  the  constant  terms  in  the  equations  of  the 
circles.  35.  3(x2  +  y2)  _  6x  -  By  =  0,  1:V^^.  36.  Imaginary.  37.  1 :  \/3. 
39.  2y  =  ±xV5,  and  2y±x\/77  +  24  =  0.  42.  4(x2  +  y2)_29x+ 12y  +  25  =  0. 
43.  2  X  +  y  =  1.  44.  The  circle  (x^  +  y2)  (\2  _  1)  _  X2(2  ax  -  a2  +  r2)  =  0,  where 
(0,  0)  is  the  fixed  point,  X  the  constant  ratio,  and  (x  —  a)2  +  y2  _  ^2  jg  t^e  fixed 
circle.  The  locus  will  be  a  straight  line  if  X  =  1.  47.  (2,  0),  (5,  0).  49.  x2  +  y^ 
= 2  r2.   50.  x2 + y2 = y2  csc2  §  •    53.  The  radius  of  the  circle  is  J  (a  +  6  ±  VoHP)  . 


ANSWERS  257 


CHAPTER  X 

Page  150.-1.  (^±fl%|-:  =  l,  ^:  +  g±'-^  =  0,  (-^%(l4^^  =  l. 
2.  It  is  perpendicular  to  the  polar  of  the  point  (x',  y')  with  respect  to  the  conic. 

Page  151.— 3.  Four. 

Page  153.  —  1.  In  the  case  of  the  ellipse  the  reflected  rays  would  converge 
and  meet  in  the  other  focus.  In  the  hyperbola  the  reflected  rays  would  diverge^ 
taking  the  directions  of  lines  which  meet  in  the  other  focus.  2.  ae^,  0,  a(e'*— 1); 

Page  154.— 1.  |V2,  (±\/2,  0),  2.      2.  ^\/l3,  (iVlS,  0),  f._    3.  ^VS,      . 
(0,  ±  V6),  V2.    4.  2,  (±2V3,  0),  6V3.    6.  i,  (l±iV3,  -2),  ^VS.    6.  jV?, 

(-1,   l±jV21),iV3.       7.^  +  ^  =  1.       8-    i  +  f  =  1-       9.  x^-2/^  =  8. 
10.   JV3,  a;2+42/2=rt2.       n.  3a;2-i/2=i2.       12.  3 xH 5 2/2= 32  ;  3^2-7x2=20. 

13.  i,  3x2  +  42/2  =  3a2.  2,  3 a;2 ^ 2,2 _ 3 ^2.      17.  y  =  x±2y/b,  y  =  xV3±2Vl3. 
Page  161.  —  1.  (-  3,  -  V3),  210?.     6.  a;2  +  ^2  =  ^2.     7.  The  locus  of  §  is  a 

circle  with  centre  at  the  other  focus  and  radius  2  a. 

Page  167.  — 1.  8x  +  27  2/  =  0.    2.   5a;  +  8y  +  30  =  0.     3.   y  =  a;  +  3. 

Page  169  —  1.  h'^x-\-  ahj  =  0, 62a;  _  a^y  =  0,  6'ic  +  a^y  =  0,  6a;  +  ay  =  0,  where 
62^2  +  a2?/2  =  a262  is  the  conic.  2.  x-y  =  0  and  a;  +  2y  =  0.  3.  32«  +  9y  =  0. 
4.   x  +  3y  =  0and  12x-25y  =  0.     6.   2»  +  3y  =  5. 

Page  175.-3.  (1)  tan-i-,  (2)  tan-i^-^  (3)  46°.  6.  6x  +  aei/  =  0and 
6ex-ay  =  0.         6.   2x  +  3y  =  0  and  6x  +  5y=[0.         13.   y  =  ±x±  Va^ ±  h\ 

[    ,  »     ^-         ).    16. ^  where  Q  is  the  angle  the  chord 

VVa2±62    V^-j-62y  a2sin2e±62cos2^^ 

makes  with  the  axis  of  the  conic.    40.   An  ellipse  with  major  axis  equal  to  the 

semi-major  axis,  and  minor  axis  equal  to  the  semi-minor  axis  of  the  given  ellipse. 

CHAPTER  XI 

Page  187.  —The  standard  forms  of  these  equations  may  be  written  as 
follows  : 
1.   t^-^x.    2.   y2  =  3x.    3.   f +  f  =  1.    4.   f-f  =  1.     6.   f-^;  =  l. 

94  6^36  468  Vl3 

10.   _^+_J^  =  4.    11.   ^-1^  =  1.    12.  y2  =  J_a;.    13.  (x-3y-l)2=0.   ^ 
5  +  V6     h-Vh  2      3  V2 

14.  ya=__2_x.       15.  — 2L_  +  _J^  =  2.       16.   y2 _ a;a -. 40,  or  xy  =  - 20. 

\/6  5-\/2     5  +  V2 


258  ANSWERS 

17.    ?!  +  y!  =  i.      18.    ^-t.  =  l,     19.    9 w2- 16x2  =  202.      20.    (5x-22/-2) 
4       1  9       4 

(5x-2y+3)=0.    21.    —^ ?^  =  1.    22.    (2a:-3y+l)(x+2?/-3)=0. 

V85+2      V85-2     2 
23.    2/2__Lx.  24.    (x  +  y  +  l)(cc-2y-2)=0.  25.   ^2_a;2  =  ioV2. 

26.  2/2  =  ^x.     27.  x2  =  3y. 
13^ 

EXAMPLES  ON  LOCI 

Page  188.  —  2.  A  parabola  whose  focus  is  the  centre  of  the  fixed  circle. 
3.  A  hyperbola,  an  ellipse,  or  a  circle.  4.  A  circle  having  the  line  joining  the 
fixed  point  and  the  centre  of  the  given  circle  for  a  diameter.  5.  An  ellipse 
whose  axes  are  the  fixed  rods.  6.  A  hyperbola,  with  one  focus  at  the  rifle  and 
the  other  at  the  target.  7.  The  two  circles  p  =  ±  r  sin  ^,  where  r  is  the  radius 
of  the  given  circle,  and  6  is  the  angle  BOG.  8.  A  circle  with  its  centre  on  the 
line  AB.  9.  A  rectangular  hyperbola  with  centre  at  0.  10.  A  circle  passing 
through  the  points  B  and  C.  11.  Two  circles  passing  through  the  points  A  and 
B,  and  having  their  centres  on  the  given  circle.  12.  A  circle  passing  through  0. 
14.  A  circle  with  centre  at  the  centre  of  the  given  triangle.  15.  A  circle  tangent 
to  the  two  equal  sides  of  the  triangle  at  the  ends  of  the  base. 

MISCELLANEOUS  PROBLEMS  ON  LOCI 

2.  A  sinusoid.  3.  A  sinusoid.  4.  This  problem  would  be  the  same  as 
No.  2,  if  the  cylinder  in  No.  2  were  an  elliptic  cylinder.  5.  Let  a  =  the 

distance  the  fly  crawls  in  a  unit  of  time,  w  =  the  angle  through  which  the 
wheel  turns  in  a  unit  of  time,  and  t  =  the  time.     Then  p  =  at  and  6  ■=  wt. 

Hence  the  polar  equation  of  the  locus  is  p  =  ( -  )  ^.     If  w  represents  the  number 

\w/  f   a  \ 

of  revolutions  the  wheel  makes  in  a  unit  of  time,  the  equation  is  p  =  r —  ^• 

\£i  TTW/ 

6.   A  series  of  parallel  lines.      7.  A  sinusoid.      12.  A  rectangular  hyperbola. 

14.  A  series  of  confocal  hyperbolas  with  foci  at  the  centres  of  the  waves. 

15.  The  locus  in  the  plane  is  a  circle.  In  space  the  locus  is  a  sphere.  16.  A 
parabola  with  its  axis  vertical.  18,  19,  20,  21.  If  the  axes  are  rectangular, 
these  curves  are  all  rectangular  hyperbolas.  23.  a  =p{\  +  ry,  where  a  repre- 
sents the  amount,  p  the  principal,  r  the  rate  per  annum,  t  the  number  of  years. 
If  the  interest  is  calculated  at  n  equal  intervals  each  year  and  added  to  principal 

as  soon  as  it  is  earned,  this  equation  becomes  a=j9  ( 1  +  -  1  •  If  we  put  n  =  mr, 
so  that  when  n  becomes  infinite  m  also  becomes  infinite,  the  equation  may  be 
written  a=j9flH —  j  *"•  If  now  n  becomes  infinite,  we  approach  the  con- 
dition in  which  the  interest  is  added  on  continuously.  The  equation  then 
becomes  a  =i)e'*,  where  e  is  the  base  of  Naperian  logarithms.  This  is  known 
as  the  Compound  Interest  Law. 


ANSWERS  259 


SOLID   GEOMETRY 

CHAPTER  XII 

Page  194. —1.   x  =  0,  ?/  =  0,  z  =  0.     z  =  0,  y  =  0;   x  =  0,  z  =  0;    x  =  0, 
y  =  0.        2.   The  planes  bisectiug  the  angles  between  the  coordinate  planes. 

3.  The  lines  through  the  origin   equally   inclined   to   the   coordinate   axes. 
6.  bx  =  ay,  cy  =  bz,  az  =  ex. 

Page  199.  — 1.    ±-^,    ±-^,    ±-^.       2.   45°  or  135°.      3.  60°  or  120°. 
V3        \/3        V3 

4.  0,  m,  n,  where  m^  +  n^  =  1.     Z,  0,  w,  where  l^  -{■  n^  =  l.     I,  m,  0,  where 

Z2  +  to2  =  1.     6.  1,0,0;  0,1,0;  0,0,1.     6.    -|=,   - -|=, i=.     -  h  h  ^^ 

Vli       Vli       Vli 

^    xi  -X2 yi  -y2 

y/ixi  -  x^y  +  iyi  -  2/2)2  +  (^;si  -  z.)^'    Vixi  -  x^y  +  {y^  -  y^y  +  (^i  -  z^Y 

Z\  —  Zi 


V(Xl  -  X2)2  +  (2/1  -  2/2)2  +  (;^j  _  ;j2)2 

CHAPTER  XIII 

Page  203.  —  1.  Straight  lines.  2.  Circles.  3.  The  ccy-contour  is  the  circle 
jp2  ^  y2  _  (.2  .  the  other  contours  are  all  straight  lines  parallel  to  the  ^-axis.  The 
locus  is  a  circular  cylinder  around  the  2!-axis. 

Page  207.  —4.  (1)  The  sphere  x^-\-y'^-\-  z'^=c'^-a'^ ;  (2)  the  plane  2ax  =  c^. 

7.  A  sphere  with  centre  at  the  centre  of  the  cube.     8.   If  the  ellipse  —  +  ^  =  1 

a^     b^ 
is  revolved  about  its  major  axis,  the  equation  of  the  generated  surface  is 

—  4-  ^  +—  =  1,  if  revolved  about  the  minor  axis  the  equation  is—  +  ^  +  —  =  1. 

The  equations  of  the  surfaces  generated  by  the  hyperbola  ^  —  ^  =  1  are 
- —  V-  —  —  =:\  and  —  -f  ^^ —  —  =  1.    The  equation  of  the  surface  generated  by 

02       62       ^,2  «2^a2       52  ^  ^  ^ 

revolving  the  parabola  ?/2  =  4  ax  about  its  axis  is  y2  _[.  ^^a  _  4  q^^.  9.  The  equa- 
tion of  the  surface  generated  by  revolving  the  parabola  z^  =  4iax  around  the 
2;-axis  is  16  oP'  (x2  +  y^)  =  2*.         10.  The  paraboloid  of  revolution  y^  +  z^=. 

lai^-a).     U.  (l)^  +  |  +  |?  =  l,where6^  =  a»-A      (2)  |-|!-|?  =  1,. 

where  62  =  c2  _  ^2.  12.  y  =  xtan  az,  where  a  depends  upon  the  number  of 
turns  the  blade  makes  per  unit  of  length. 

CHAPTER  XTV 

Page  214.  —  The  symmetric  equations  may  be  written  as  follows  : 
2    ^-y-S-g-2  3    a;_y-4_g-3  .    x  +  2_y _z-l 

"2-5-3'  12  3    '  '31       -6* 


260 


ANSWERS 


5.    ?  =  yZlM: 

2  3 


M. 


0 


6. 


a  line  perpendicular  to  the  2!-axis; 

the  plane  e  =  0.  8.  «  —  2  =  y  +  3  = 

10.  ?  =  1^  =  ^.       11.    (0,6,3). 

70°  32'. 
8.  ±-^-1  =  ^-^1^  =  =^:^.      10.  cos-i-^ 


1        1        -1 

X  —  a  _y  —  b  _z 
0     ~     0     ~ 


a_y  -  h 
m 


z  —  c 
~0~' 


z-\. 


9. 


—  c 

x-4- 1 


?  =  1^  =  -. 
Z      m     n 

Page  216. — 3.   cos-^|  = 
ar  —  l_y  —  4_g  —  3 
3 


4.   60°. 
:  109°  28'. 


3 


5.  cos 


perpendicular  to 

y-3_g--2 
-6        -2' 

3 


35°  16^ 


2  4 

Page  218.  —  8.  A  plane.  13.  If  the  equations  of  the  two  fixed  lines  are 
y  +  mx  =  0,  2;  +  c  =  0,  and  y  —  mx  =  0,_  2;  -  c  =  0  (§  147),  the  equation  of  the 
locus  is  2/2  _  wi%2  _  (^a  _  1)  (2;2  -  c2). 

14. .  Let  the  equations  of  the  two  fixed  lines  be  taken  as  in  Ex.  13,  and  let  2 1 
be  the  constant  length  of  the  moving  line.  Then  the  equation  of  the  surface  is 
c\cy  -  mxzy^  +  m\\cmx  -  yzY  +  m\c^  -  P)  (c^  -  z'^Y  =  0.  The  locus  de- 
scribed by  any  fixed  point  on  the  line  is  found  by  putting  z  =  k\n  this  equation, 
where  k  is  any  constant. 

16.  If  the  fixed  lines  are  the  same  as  in  Ex.  13,  the  equation  of  the  surface 
is  y2!  —  cmx  =  0. 

CHAPTER  XV 

Page  222.  —  1.  The  constant  term  D  represents  the  square  of  the  length  of 
any  tangent  drawn  from  the  origin  to  the  sphere,  or  the  product  of  the  segmei4% 
into  which  the  origin  divides  any  chord  passing  through  the  origin.  The  origifi 
is  outside,  on,  or  inside  the  sphere,  according  as  D  is  positive,  zero,  or  negative. 
If  nn  =  —  d2,  P  is  at  the  centre  of  the  sphere. 

2.  Four,  since  there  are  four  independent  constants  in  the  general  equation. 

8.   The  equation  of  a  sphere  through  the  four  points  (Xi,  2/1,  zi),  (xa,  2/2,  «2)» 
(«3,  ys,  «8),  («4, 2/4,  «4)  may  be  written : 
+  2/2  +z^ 


X,     J/,     z,    1 
Xi2  +  2/i2  +  012,  xx,  yu  «i,  1 
1 
1 

1 


Xi^  +  2/2^  +  ^2%    «2,    2/2,    ^2, 


=  0. 


Xs^  +  ys^  +  zs»,  xs,  2/3,  Z8, 
Xi^  +  yi^  +  Zi%  X4,  2/4,  «4, 

4.   (1,  2,  3),  r  =  3,  outside,  t  =  VE;  intercepts  1  ±  2V^  ;  2  ±  V^ ;  1,  5. 

6.   (-  6, 12,  0),  r  =  13,  on,  «  =  0 ;  intercepts  0,  -  10  ;  0,  24  ;  0,  0. 

6.  (-  3, 4,  -  1),  r  =  6,  inside,  t  =  V-  10  ;  intercepts  -3  ±  VTd ;  4  ±  V26 ; 
-1±  VTT.  7.  (2,  -3,  -5),  r=  \/38,  on,  t=0  ;  intercepts  0,  4  ;  0,  -6  ;  0,  -10. 
8.  x^  +  y^  +  z^±2rx  =  0.  9.  (x  ±  r)2+ (2/ ±  r)2+ (^  -  c)2  =  r2. 

10.  (x±r)2+(2/±r)2+(«±r)2=r8.  -  H.  x2+2/H«2±2ax±2a2/±2c«+a2=0. 
12.  x2+2/2+«2^2ax±2a2^±2a«+a2=0.  13.  x2+2/2+«2-4x-42/±4«+4=0, 
and  x2+y2+«2-20x-202/±20«+100=0. 


ANSWERS  261 

19.  The  radical  axis  of  three  spheres  is  the  locus  of  all  points  from  which 
tangents  drawn  to  the  spheres  are  equal.  The  radical  centre  of  four  spheres  is 
the  point  from  which  taugents  drawn  to  the  four  spheres  are  all  equal.  If  2)i, 
2>2,  i>3,  2>4  are  the  constant  terms  in  the  equations  of  the  circles,  the  condition 
isDi=2>2=^3  =  2>4. 

Pag©  237.  —  3.  Sections  of  the  two  surfaces  are  equal,  if  their  planes  are 
parallel  to  the  y^-plane,  and  the  difference  of  the  squares  of  their  distance*;  fr^m 
*^s^ plane  is  equal  to  a^.  '.  •_  .  " :.  ,  J'-j-ttii^f 

^^,<y%'  In  both  cases  the  asymptotic  conis  pas/ses  entirely  itifo.-^i^fcfi^ijt^glwK 
^    Page  243.  — 6.  a;2+2/H«2=a2.-     11.  "^^^^.^.(y^^^ 
The  area  of  the  section  .made  by  thft  pJaHe'^'^ii  isindie'^)^!^  r,1[)e^&!^'a 

variation  in  r  causes  no  chaoge  in  th^^s^io^vj^^sgltjer^v  or^bf  'ite'distan!i;e'|r^m 
the  y5;-plane ;  it  produces  athanee  on>pKi(^BJK^j(icioB3inate  ofthe  centj^.*-  - 

^^  /  APPENDIX  '^^?' 

Page  246.  —  1,  2,  8.  The  same  a^the  ayfxis.  4.  The  y-axis.  6..  The  je-axis. 
6.  The  lines  y  =  x  and  2^=  —  «,  -fei^i^^O, 0)  y  =  2a;,  at  (1,  0)  y  =-J  x:-}- 1, 
at  (2,  0)  y  =  2  X  -  4.  /  8.  At  ,(()^:%V  ==  0,  at  (1,  oj  2^  =  2  x  -  2,  at  (^  1,  0) 
2/  =-  2x~  2.  9.,.At  (0,  0)  V=5^'l5!^,.at  (-  2,  0)  y  =  lOx  -  20,  at  (3,  0) 

2/=Jl^x-45.  10.  A:t.(.Q,  12,)'yi:^r^ilx  +  12,  at  (1,  0)  y  =  -  12:^'il), 

at(--8,())y  =  28(x+5),  at(4,0)2/  =  21(x-4).  '  .S"^*" 


,'^'-  A*^''^ 

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